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%I A330892 #43 Dec 19 2024 11:27:31
%S A330892 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,
%T A330892 -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,
%U A330892 -80,-27,8,28,36,35,28,18,8,1,0,-99,-35,9,36,49,51,45,34,21,9,1,0
%N A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).
%C A330892 \c  0 1  2  3  4   5   6   7   8   9  10  11   12   13    14  15  ...
%C A330892 r\
%C A330892 _0  0 1  0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195  A067998
%C A330892 _1  0 1  1  0 -2  -5  -9 -14 -20 -27 -35 -44  -54  -65  -77  -90  A080956
%C A330892 _2  0 1  2  3  4   5   6   7   8   9  10  11   12   13   14   15  A001477
%C A330892 _3  0 1  3  6 10  15  21  28  36  45  55  66   78   91  105  120  A000217
%C A330892 _4  0 1  4  9 16  25  36  49  64  81 100 121  144  169  196  225  A000290
%C A330892 _5  0 1  5 12 22  35  51  70  92 117 145 176  210  247  287  330  A000326
%C A330892 _6  0 1  6 15 28  45  66  91 120 153 190 231  276  325  378  435  A000384
%C A330892 _7  0 1  7 18 34  55  81 112 148 189 235 286  342  403  469  540  A000566
%C A330892 _8  0 1  8 21 40  65  96 133 176 225 280 341  408  481  560  645  A000567
%C A330892 _9  0 1  9 24 46  75 111 154 204 261 325 396  474  559  651  750  A001106
%C A330892 10  0 1 10 27 52  85 126 175 232 297 370 451  540  637  742  855  A001107
%C A330892 11  0 1 11 30 58  95 141 196 260 333 415 506  606  715  833  960  A051682
%C A330892 12  0 1 12 33 64 105 156 217 288 369 460 561  672  793  924 1065  A051624
%C A330892 13  0 1 13 36 70 115 171 238 316 405 505 616  738  871 1015 1170  A051865
%C A330892 14  0 1 14 39 76 125 186 259 344 441 550 671  804  949 1106 1275  A051866
%C A330892 15  0 1 15 42 82 135 201 280 372 477 595 726  870 1027 1197 1380  A051867
%C A330892 ...
%C A330892 Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.
%H A330892 E. Deza and M. Deza, <a href="/load/view.php?a=aHR0cDovL3d3dy53b3JsZHNjaWVudGlmaWMuY29tL2RvaS9zdXBwbC8xMC4xMTQyLzgxODgvc3VwcGxfZmlsZS84MTg4X2NoYXAwMS5wZGY">Figurate Numbers</a>, World Scientific, 2012; see p. 45.
%H A330892 Eric Weisstein's World of Mathematics, <a href="/load/view.php?a=aHR0cHM6Ly9tYXRod29ybGQud29sZnJhbS5jb20vUG9seWdvbmFsTnVtYmVyLmh0bWw">Polygonal Number</a>.
%H A330892 Wikipedia, <a href="/load/view.php?a=aHR0cHM6Ly9lbi53aWtpcGVkaWEub3JnL3dpa2kvUG9seWdvbmFsX251bWJlcg">Polygonal number</a>.
%H A330892 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1BvbCNwb2x5Z29uYWxfbnVtYmVycw">Index to sequences related to polygonal numbers</a>
%F A330892 P(r, c) = (r - 2)(c(c-1)/2) + c.
%t A330892 Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten
%Y A330892 Cf. A317302 (the same array) but read by ascending antidiagonals.
%Y A330892 Sub-arrays: A089000, A139600, A206735;
%Y A330892 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, ..., ;
%Y A330892 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;
%Y A330892 Diagonals: A256857, A127736, A002411, A006003, A006000, A064808, A060354, A162607, A077414,
%Y A330892 Number of times k>1 appears: A129654, First occurrence of k: A063778.
%K A330892 easy,sign,tabl
%O A330892 1,7
%A A330892 _Robert G. Wilson v_, Apr 27 2020

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