OFFSET
0,6
LINKS
G. C. Greubel, Antidiagonal rows n = 0..25, flattened
FORMULA
T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ((k+1)^5 - (k+1)^4 - (k+1)^3 + (k+1)^2)^i ) with T(n, 0) = n! (square array).
T(n, k) = ( Product_{j=1..n} ((k^2*(k+1)^2*(k+2))^j -1) )/(k^2*(k+1)^2*(k+2) -1)^n with T(n, 0) = n! (square array). - G. C. Greubel, Jun 14 2021
EXAMPLE
Square array begins as:
1, 1, 1, 1, ...;
1, 1, 1, 1, ...;
2, 13, 145, 721, ...;
6, 2041, 3027745, 374286241, ...;
24, 3847285, 9104020469425, 139895890728482161, ...;
Antidiagonal triangle begins as:
1;
1, 1;
1, 1, 2;
1, 1, 13, 6;
1, 1, 145, 2041, 24;
1, 1, 721, 3027745, 3847285, 120;
1, 1, 2401, 374286241, 9104020469425, 87029433985, 720; ...
MATHEMATICA
(* First program *)
T[n_, m_] = If[m==0, n!, Product[Sum[((m+1)^5 -(m+1)^4 -(m+1)^3 +(m+1)^2)^i, {i, 0, k-1}], {k, n}]];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 14 2021 *)
(* Second program *)
T[n_, k_]= If[k==0, n!, Product[((k^2*(k+1)^2*(k+2))^j -1), {j, n}]/(k^2*(k+1)^2*(k+2) -1)^n];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 12 2021 *)
PROG
(Sage)
def A156885(n, k): return factorial(n) if (k==0) else product(( (k^2*(k+1)^2*(k+2))^j -1) for j in (1..n))/(k^2*(k+1)^2*(k+2) -1)^n
flatten([[A156885(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 17 2009
EXTENSIONS
Edited by Joerg Arndt and G. C. Greubel, Jun 14 2021
STATUS
approved