%I #45 May 08 2022 22:48:17

%S 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 1,7,18,34,55,81,112,148,1,8,21,40,65,96,133,176,225,1,9,24,46,75,111,

%U 154,204,261,325,1,10,27,52,85,126,175,232,297,370,451

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

%H T. D. Noe, <a href="/A081422/b081422.txt">Rows n = 0..100 of triangle, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygonalNumber.html">Polygonal Number</a>

%F 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 T(n, k) = k*((n-2)*k -(n-4))/2 (see MathWorld link). - _Michel Marcus_, Jun 22 2015

%e The array starts

%e 1 1 3 10 ...

%e 1 2 6 16 ...

%e 1 3 9 22 ...

%e 1 4 12 28 ...

%e The triangle starts

%e 1;

%e 1, 1;

%e 1, 2, 3;

%e 1, 3, 6, 10;

%e 1, 4, 9, 16, 25;

%e ...

%t 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 (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 (Magma) [[k*((n-2)*k-(n-4))/2: k in [1..n+1]]: n in [0..10]]; // _G. C. Greubel_, Oct 13 2018

%o (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 (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 Rows include A060354, A064808, A006000, A006003, A002411.

%Y Diagonals include A001093, A053698, A069778, A000578, A002414, A081423, A081435, A081436, A081437, A081438, A081441.

%Y Antidiagonals are composed of n-gonal numbers.

%K easy,nonn,tabl,look

%O 0,5

%A _Paul Barry_, Mar 21 2003