The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A139601

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A139601

Square array of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)*(k - 1)*k/2 + k.
(history; published version)

#35 by Peter Luschny at Sat Jul 13 09:39:10 EDT 2024

STATUS

proposed

approved

#34 by Peter Luschny at Sat Jul 13 02:17:56 EDT 2024

STATUS

editing

proposed

#33 by Peter Luschny at Sat Jul 13 02:17:40 EDT 2024

NAME

Square array T(n,k) = (n+1)*(k-1)*k/2+k, of polygonal numbers, read by ascending antidiagonals upwards: T(n, k) = (n + 1)*(k - 1)*k/2 + k.

STATUS

proposed

editing

#32 by G. C. Greubel at Sat Jul 13 02:00:22 EDT 2024

STATUS

editing

proposed

#31 by G. C. Greubel at Sat Jul 13 01:58:54 EDT 2024

FORMULA

t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2) , where t(antidiagonal trianglen,k) is this array read by rising antidiagonals.

STATUS

proposed

editing

#30 by G. C. Greubel at Sat Jul 13 00:24:35 EDT 2024

STATUS

editing

proposed

Discussion

Sat Jul 13

01:47

Joerg Arndt: "antidiagonal triangle" -->
"where t(n,k) is this array read by falling/rising(?) antidiagonals" ?

#29 by G. C. Greubel at Sat Jul 13 00:24:25 EDT 2024

FORMULA

t(2*n, n) = A006003(n).

t(2*n+1, n) = A002411(n).

t(2*n-1, n) = A006000(n-1).

Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).

Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)

CROSSREFS

Cf. A000007, A000012, A000027, A002411, A006000, A006003, A006522.

Cf. A008585, A016957, A017329, A086270, A117142, A139606, A139607.

Cf. A000007, A000012, A000027, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A057145, A086271, A139600.

Cf. A086270, A139615, A139616, A057145, A086271, A139600, A139617, A139618, A139619, A139620.

Cf. A006522, A117142A139619, A139620.

#28 by G. C. Greubel at Fri Jul 12 23:58:02 EDT 2024

FORMULA

T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009

From G. C. Greubel, Jul 12 2024: (Start)

t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2) (antidiagonal triangle).

Sum_{k=0..n} t(n, k) = A006522(n+2).

Sum_{k=0..n} (-1)^k*t(n, k) = A117142().

PROG

(Magma)

T:= func< n, k | k*((n+1)*(k-1) +2)/2 >;

A139601:= func< n, k | T(n-k, k) >;

[A139601(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024

(SageMath)

def T(n, k): return k*((n+1)*(k-1)+2)/2

def A139601(n, k): return T(n-k, k)

flatten([[A139601(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024

CROSSREFS

Cf. A006522, A117142

STATUS

approved

editing

#27 by Michael De Vlieger at Thu Mar 07 11:13:09 EST 2024

STATUS

proposed

approved

#26 by Michel Marcus at Thu Mar 07 11:08:04 EST 2024

STATUS

editing

proposed