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A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.

%I A081422 #46 Feb 16 2025 08:32:48
%S A081422 1,1,1,1,2,3,1,3,6,10,1,4,9,16,25,1,5,12,22,35,51,1,6,15,28,45,66,91,
%T A081422 1,7,18,34,55,81,112,148,1,8,21,40,65,96,133,176,225,1,9,24,46,75,111,
%U A081422 154,204,261,325,1,10,27,52,85,126,175,232,297,370,451
%N A081422 Triangle read by rows in which row n consists of the first n+1 n-gonal numbers.
%H A081422 T. D. Noe, <a href="/A081422/b081422.txt">Rows n = 0..100 of triangle, flattened</a>
%H A081422 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>
%F A081422 Array of coefficients of x in the expansions of T(k, x) = (1 + k*x -(k-2)*x^2)/(1-x)^4, k > -4.
%F A081422 T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - _Michel Marcus_, Jun 22 2015
%e A081422 The array starts
%e A081422   1  1  3 10 ...
%e A081422   1  2  6 16 ...
%e A081422   1  3  9 22 ...
%e A081422   1  4 12 28 ...
%e A081422 The triangle starts
%e A081422   1;
%e A081422   1,  1;
%e A081422   1,  2,  3;
%e A081422   1,  3,  6, 10;
%e A081422   1,  4,  9, 16, 25;
%e A081422   ...
%t A081422 Table[PolygonalNumber[n,i],{n,0,10},{i,n+1}]//Flatten (* Requires Mathematica version 10.4 or later *) (* _Harvey P. Dale_, Aug 27 2016 *)
%o A081422 (PARI) tabl(nn) = {for (n=0, nn, for (k=1, n+1, print1(k*((n-2)*k-(n-4))/2, ", ");); print(););} \\ _Michel Marcus_, Jun 22 2015
%o A081422 (Magma) [[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // _G. C. Greubel_, Oct 13 2018
%o A081422 (Sage) [[k*((n-2)*k -(n-4))/2 for k in (1..n+1)] for n in (0..10)] # _G. C. Greubel_, Aug 14 2019
%o A081422 (GAP) Flat(List([0..10], n-> List([1..n+1], k-> k*((n-2)*k-(n-4))/2 ))); # _G. C. Greubel_, Aug 14 2019
%Y A081422 Rows include A060354, A064808, A006000, A006003, A002411.
%Y A081422 Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441.
%Y A081422 Antidiagonals are composed of n-gonal numbers.
%K A081422 easy,nonn,tabl,look
%O A081422 0,5
%A A081422 _Paul Barry_, Mar 21 2003