%I #19 Sep 24 2021 16:35:08

%S 1,1,1,1,9,8,1,36,251,216,1,100,2555,16280,13824,1,225,15055,335655,

%T 2048824,1728000,1,441,63655,3587535,74550304,444273984,373248000,1,

%U 784,214918,25421200,1305074809,26015028256,152759224512,128024064000,1,1296,616326,135459216,14320729209,694213330464,13472453691584,78340747014144,65548320768000

%N Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^3*x).

%C Column 1 forms the squares of the triangular numbers (A000537).

%C Main diagonal forms the cubes of the factorial numbers (A000442).

%C Row sums equal Product_{k=1..n} (k^3 + 1) = n!*Product_{k=1..n} (k*(k-1) + 1) = n!*A130032(n).

%H Seiichi Manyama, <a href="/A249677/b249677.txt">Rows n = 0..139, flattened</a>

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 9, 8;

%e 1, 36, 251, 216;

%e 1, 100, 2555, 16280, 13824;

%e 1, 225, 15055, 335655, 2048824, 1728000;

%e 1, 441, 63655, 3587535, 74550304, 444273984, 373248000;

%e 1, 784, 214918, 25421200, 1305074809, 26015028256, 152759224512, 128024064000;

%e 1, 1296, 616326, 135459216, 14320729209, 694213330464, 13472453691584, 78340747014144, 65548320768000; ...

%o (PARI) {T(n,k)=polcoeff(prod(m=0,n,1 + m^3*x +x*O(x^n)),k)}

%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

%Y Cf. A000537, A000442, A008955, A107415, A130032.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Nov 03 2014