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)^3 - (k+1))^i ) with T(n, 0) = n! (square array).
T(n, k) = ( Product_{j=1..n} ((k*(k+1)*(k+2))^j -1) )/(k*(k+1)*(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, 1, 1, 1, 1 ...;
2, 7, 25, 61, 121, 211 ...;
6, 301, 15025, 223321, 1757041, 9349621 ...;
24, 77959, 216735625, 49054914181, 3061680840361, 87001131137131 ...;
Triangle begins as:
1;
1, 1;
1, 1, 2;
1, 1, 7, 6;
1, 1, 25, 301, 24;
1, 1, 61, 15025, 77959, 120;
1, 1, 121, 223321, 216735625, 121226245, 720;
1, 1, 211, 1757041, 49054914181, 75034090110625, 1131162092095, 5040;
MATHEMATICA
(* First program *)
T[n_, m_] = If[m==0, n!, Product[Sum[(-(m+1) + (m+1)^3)^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 12 2021 *)
(* Second program *)
T[n_, k_]= If[k==0, n!, Product[((k*(k+1)*(k+2))^j -1), {j, n}]/(k*(k+1)*(k+2) - 1)^n];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 14 2021 *)
PROG
(Sage)
def A156882(n, k): return factorial(n) if (k==0) else product(((k*(k+1)*(k+2))^j -1) for j in (1..n))/(k*(k+1)*(k+2)-1)^n
flatten([[A156882(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 G. C. Greubel, Jun 14 2021
STATUS
approved