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Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).
+20
0
0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0
COMMENTS
\c 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
r\
_0 0 1 0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195 A067998_1 0 1 1 0 -2 -5 -9 -14 -20 -27 -35 -44 -54 -65 -77 -90 A080956_2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 A001477_3 0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 A000217_4 0 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 A000290_5 0 1 5 12 22 35 51 70 92 117 145 176 210 247 287 330 A000326_6 0 1 6 15 28 45 66 91 120 153 190 231 276 325 378 435 A000384_7 0 1 7 18 34 55 81 112 148 189 235 286 342 403 469 540 A000566_8 0 1 8 21 40 65 96 133 176 225 280 341 408 481 560 645 A000567_9 0 1 9 24 46 75 111 154 204 261 325 396 474 559 651 750 A00110610 0 1 10 27 52 85 126 175 232 297 370 451 540 637 742 855 A00110711 0 1 11 30 58 95 141 196 260 333 415 506 606 715 833 960 A05168212 0 1 12 33 64 105 156 217 288 369 460 561 672 793 924 1065 A05162413 0 1 13 36 70 115 171 238 316 405 505 616 738 871 1015 1170 A05186514 0 1 14 39 76 125 186 259 344 441 550 671 804 949 1106 1275 A05186615 0 1 15 42 82 135 201 280 372 477 595 726 870 1027 1197 1380 A051867...
Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.
FORMULA
P(r, c) = (r - 2)(c(c-1)/2) + c.
MATHEMATICA
Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten
CROSSREFS
Cf. A317302 (the same array) but read by ascending antidiagonals. Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ; Columns (maybe missing some leading terms): A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;
Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.
+10
162
1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1
COMMENTS
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section). The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270. The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n. Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels. Odd-indexed rows are also the rows of the irregular triangle A237270. Even-indexed rows are also the rows of the triangle A237593. Lengths of the odd-indexed rows are in A237271. Lengths of the even-indexed rows give 2* A003056. Row sums of the odd-indexed rows gives A000203, the sum of divisors function. Row sums of the even-indexed rows give the positive even numbers (see A005843). From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
EXAMPLE
Irregular triangle begins:
1;
1, 1;
3;
2, 2;
2, 2;
2, 1, 1, 2;
7;
3, 1, 1, 3;
3, 3;
3, 2, 2, 3;
12;
4, 1, 1, 1, 1, 4;
4, 4;
4, 2, 1, 1, 2, 4;
15;
5, 2, 1, 1, 2, 5;
5, 3, 5;
5, 2, 2, 2, 2, 5;
9, 9;
6, 2, 1, 1, 1, 1, 2, 6;
6, 6;
6, 3, 1, 1, 1, 1, 3, 6;
28;
7, 2, 2, 1, 1, 2, 2, 7;
7, 7;
7, 3, 2, 1, 1, 2, 3, 7;
12, 12;
8, 3, 1, 2, 2, 1, 3, 8;
8, 8, 8;
8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
31;
9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
1 1 = 1 |_| | | | | | | | | | | | | | | |
2 3 = 3 |_ _|_| | | | | | | | | | | | | |
3 4 = 2 + 2 |_ _| _|_| | | | | | | | | | | |
4 7 = 7 |_ _ _| _|_| | | | | | | | | |
5 6 = 3 + 3 |_ _ _| _| _ _|_| | | | | | | |
6 12 = 12 |_ _ _ _| _| | _ _|_| | | | | |
7 8 = 4 + 4 |_ _ _ _| |_ _|_| _ _|_| | | |
8 15 = 15 |_ _ _ _ _| _| | _ _ _|_| |
9 13 = 5 + 3 + 5 |_ _ _ _ _| | _|_| | _ _ _|
10 18 = 9 + 9 |_ _ _ _ _ _| _ _| _| |
11 12 = 6 + 6 |_ _ _ _ _ _| | _| _| _|
12 28 = 28 |_ _ _ _ _ _ _| |_ _| _|
13 14 = 7 + 7 |_ _ _ _ _ _ _| | _ _|
14 24 = 12 + 12 |_ _ _ _ _ _ _ _| |
15 24 = 8 + 8 + 8 |_ _ _ _ _ _ _ _| |
16 31 = 31 |_ _ _ _ _ _ _ _ _|
...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
Level _ _
1 _|1|1|_
2 _|2 _|_ 2|_
3 _|2 |1|1| 2|_
4 _|3 _|1|1|_ 3|_
5 _|3 |2 _|_ 2| 3|_
6 _|4 _|1|1|1|1|_ 4|_
7 _|4 |2 |1|1| 2| 4|_
8 _|5 _|2 _|1|1|_ 2|_ 5|_
9 _|5 |2 |2 _|_ 2| 2| 5|_
10 _|6 _|2 |1|1|1|1| 2|_ 6|_
11 _|6 |3 _|1|1|1|1|_ 3| 6|_
12 _|7 _|2 |2 |1|1| 2| 2|_ 7|_
13 _|7 |3 |2 _|1|1|_ 2| 3| 7|_
14 _|8 _|3 _|1|2 _|_ 2|1|_ 3|_ 8|_
15 _|8 |3 |2 |1|1|1|1| 2| 3| 8|_
16 |9 |3 |2 |1|1|1|1| 2| 3| 9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n. The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n). The total number of vertical line segments in the n-th level of the diagram equals A131507(n). The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - Omar E. Pol, Aug 28 2018 The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - Omar E. Pol, Nov 09 2022
CROSSREFS
Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871. Cf. A000079 (powers of 2)........., for the characteristic shape see A346872. Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level. Cf. A000217 (triangular numbers).., for the characteristic shape see A346873. Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875. Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876. Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level. Cf. A008586 (multiples of 4)......, perimeters of the successive levels. Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613. Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo. Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864. Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level. Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others). Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238. Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.
The n-th n-gonal number: a(n) = n*(n^2 - 3*n + 4)/2.
+10
37
0, 1, 2, 6, 16, 35, 66, 112, 176, 261, 370, 506, 672, 871, 1106, 1380, 1696, 2057, 2466, 2926, 3440, 4011, 4642, 5336, 6096, 6925, 7826, 8802, 9856, 10991, 12210, 13516, 14912, 16401, 17986, 19670, 21456, 23347, 25346, 27456, 29680, 32021
COMMENTS
Binomial transform of (0,1,0,3,0,0,0,...). - Paul Barry, Sep 14 2006 Also the number of permutations of length n which can be sorted by a single cut-and-paste move (in the sense of Cranston, Sudborough, and West). - Vincent Vatter, Aug 21 2013
FORMULA
a(n) = (n*(n-2)^2 + n^2)/2.
E.g.f.: exp(x)*x*(1+x^2/2). - Paul Barry, Sep 14 2006
MATHEMATICA
Table[(n (n-2)^2+n^2)/2, {n, 0, 50}] (* Harvey P. Dale, Aug 05 2011 *) CoefficientList[Series[x (1 - 2 x + 4 x^2) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Feb 16 2015 *) Table[PolygonalNumber[n, n], {n, 0, 50}] (* Harvey P. Dale, Mar 07 2016 *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 2, 6}, 50] (* Harvey P. Dale, Mar 07 2016 *)
PROG
(PARI) a(n) = { (n*(n - 2)^2 + n^2)/2 } \\ Harry J. Smith, Jul 04 2009
AUTHOR
Hareendra Yalamanchili (hyalaman(AT)mit.edu), Apr 01 2001
Array A(n, k) read by ascending antidiagonals. Polygonal number weighted generalized Catalan sequences.
+10
6
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 15, 8, 1, 1, 1, 5, 34, 105, 16, 1, 1, 1, 6, 61, 496, 945, 32, 1, 1, 1, 7, 96, 1385, 11056, 10395, 64, 1, 1, 1, 8, 139, 2976, 50521, 349504, 135135, 128, 1, 1, 1, 9, 190, 5473, 151416, 2702765, 14873104, 2027025, 256, 1
COMMENTS
Using polygonal numbers as weights, a recursion for triangles is defined, whose main diagonals represents a family of sequences, which include, among others, the powers of 2, the double factorial of odd numbers, the reduced tangent numbers, and the Euler numbers.
Apart from the edge cases k = 0 and k = n the recursion is T(n, k) = w(n, k) * T(n, k - 1) + T(n - 1, k). T(n, 0) = 1 and T(n, n) = T(n, n-1) if n > 0.
The weights w(n, k) identical to 1 yield the recursion of the Catalan triangle A009766 (with main diagonal the Catalan numbers). Here the polygonal numbers are used as weights in the form w(n, k) = p(s, n - k + 1), where the parameter s is the number of sides of the polygon and p(s, n) = ((s-2) * n^2 - (s-4) * n) / 2, see A317302.
EXAMPLE
Array A(n, k) starts: (polygon|diagonal|triangle)
[8] 1, 1, 9, 249, 14001, 1322001, 188106489, ... A000567
MAPLE
poly := (s, n) -> ((s - 2) * n^2 - (s - 4) * n) / 2:
T := proc(s, n, k) option remember; if k = 0 then 1 else if k = n then T(s, n, k-1) else poly(s, n - k + 1) * T(s, n, k - 1) + T(s, n - 1, k) fi fi end:
for n from 0 to 8 do A := (n, k) -> T(n, k, k): seq(A(n, k), k = 0..9) od;
# Alternative, using continued fractions:
A := proc(p, L) local CF, poly, k, m, P, ser;
poly := (s, n) -> ((s - 2)*n^2 - (s - 4)*n)/2;
CF := 1 + x;
for k from 1 to L do
m := L - k + 1;
P := poly(p, m);
CF := 1/(1 - P*x*CF)
od;
ser := series(CF, x, L);
seq(coeff(ser, x, m), m = 0..L-1)
end:
for p from 0 to 8 do lprint(A(p, 8)) od;
MATHEMATICA
poly[s_, n_] := ((s - 2) * n^2 - (s - 4) * n) / 2;
T[s_, n_, k_] := T[s, n, k] = If[k == 0, 1, If[k == n, T[s, n, k - 1], poly[s, n - k + 1] * T[s, n, k - 1] + T[s, n - 1, k]]];
A[n_, k_] := T[n, k, k];
Table[A[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 27 2023, from first Maple program *)
PROG
(Python)
from functools import cache
@cache
def T(s, n, k):
if k == 0: return 1
if k == n: return T(s, n, k - 1)
p = (n - k + 1) * ((s - 2) * (n - k + 1) - (s - 4)) // 2
return p * T(s, n, k - 1) + T(s, n - 1, k)
def A(n, k): return T(n, k, k)
for n in range(9): print([A(n, k) for k in range(9)])
(PARI)
A(p, n) = {
my(CF = 1 + x,
poly(s, n) = ((s - 2)*n^2 - (s - 4)*n)/2,
m, P
);
for(k = 1, n,
m = n - k + 1;
P = poly(p, m);
CF = 1/(1 - P*x*CF)
);
Vec(CF + O(x^(n)))
}
for(p = 0, 8, print(A(p, 8)))
Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m*((n - 2)*m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.
+10
5
0, 0, 1, 0, 1, -3, 0, 1, -2, 0, 0, 1, -1, 1, -8, 0, 1, 0, 2, -5, -3, 0, 1, 1, 3, -2, 0, -15, 0, 1, 2, 4, 1, 3, -9, -8, 0, 1, 3, 5, 4, 6, -3, -2, -24, 0, 1, 4, 6, 7, 9, 3, 4, -14, -15, 0, 1, 5, 7, 10, 12, 9, 10, -4, -5, -35, 0, 1, 6, 8, 13, 15, 15, 16, 6, 5, -20, -24, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, -5, -9, -48
COMMENTS
Note that the formula mentioned in the definition gives several kinds of numbers, for example:
Row 2 gives A001057 (canonical enumeration of integers). Row 3 gives 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6). Row 4 gives A008794 (squares repeated) except the initial zero. Finally, for n >= 5 row n gives the generalized k-gonal numbers (see Crossrefs section).
EXAMPLE
Array begins:
------------------------------------------------------------------
n\m Seq. No. 0 1 -1 2 -2 3 -3 4 -4 5 -5
------------------------------------------------------------------
0 A317300: 0, 1, -3, 0, -8, -3, -15, -8, -24, -15, -35... 1 A317301: 0, 1, -2, 1, -5, 0, -9, -2, -14, -5, -20... 2 A001057: 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5... 3 ( A008795): 0, 1, 0, 3, 1, 6, 3, 10, 6, 15, 10... 4 ( A008794): 0, 1, 1, 4, 4, 9, 9, 16, 16, 25, 25... 5 A001318: 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40... 6 A000217: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55... 7 A085787: 0, 1, 4, 7, 13, 18, 27, 34, 46, 55, 70... 8 A001082: 0, 1, 5, 8, 16, 21, 33, 40, 56, 65, 85... 9 A118277: 0, 1, 6, 9, 19, 24, 39, 46, 66, 75, 100... 10 A074377: 0, 1, 7, 10, 22, 27, 45, 52, 76, 85, 115... 11 A195160: 0, 1, 8, 11, 25, 30, 51, 58, 86, 95, 130... 12 A195162: 0, 1, 9, 12, 28, 33, 57, 64, 96, 105, 145... 13 A195313: 0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160... 14 A195818: 0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175... 15 A277082: 0, 1, 12, 15, 37, 42, 75, 82, 126, 135, 190... ...
MATHEMATICA
t[n_, r_] := PolygonalNumber[n, If[OddQ@ r, Floor[(r + 1)/2], -r/2]]; Table[ t[n - r, r], {n, 0, 11}, {r, 0, n}] // Flatten (* also *)
(* to view the square array *) Table[ t[n, r], {n, 0, 15}, {r, 0, 10}] // TableForm (* Robert G. Wilson v, Aug 08 2018 *)
CROSSREFS
Column 3 gives A001477 which coincides with the row numbers. Row 3 gives 0 together with A008795. For n >= 5, rows n gives the generalized n-gonal numbers: A001318 (n=5), A000217 (n=6), A085787 (n=7), A001082 (n=8), A118277 (n=9), A074377 (n=10), A195160 (n=11), A195162 (n=12), A195313 (n=13), A195818 (n=14), A277082 (n=15), A274978 (n=16), A303305 (n=17), A274979 (n=18), A303813 (n=19), A218864 (n=20), A303298 (n=21), A303299 (n=22), A303303 (n=23), A303814 (n=24), A303304 (n=25), A316724 (n=26), A316725 (n=27), A303812 (n=28), A303815 (n=29), A316729 (n=30). Cf. A317302 (a similar table but with polygonal numbers).
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