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)^2 - (k+1))^i ) with T(n, 0) = n! (square array).
T(n, k) = ( Product_{j=1..n} (k^j*(k+1)^j -1) )/(k^2 + k -1)^n with T(n, 0) = n! (square array). - G. C. Greubel, Jun 12 2021
EXAMPLE
Square array begins as:
1, 1, 1, 1, 1, ...;
1, 1, 1, 1, 1, ...;
2, 3, 7, 13, 21, ...;
6, 21, 301, 2041, 8841, ...;
24, 315, 77959, 3847285, 74450061, ...;
Triangle begins as:
1;
1, 1;
1, 1, 2;
1, 1, 3, 6;
1, 1, 7, 21, 24;
1, 1, 13, 301, 315, 120;
1, 1, 21, 2041, 77959, 9765, 720;
1, 1, 31, 8841, 3847285, 121226245, 615195, 5040;
1, 1, 43, 28861, 74450061, 87029433985, 1131162092095, 78129765, 40320;
MATHEMATICA
(* First program *)
T[n_, m_] = If[m==0, n!, Product[Sum[(-(m+1) + (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 12 2021 *)
(* Second program *)
T[n_, k_]= If[k==0, n!, Product[(k^j*(k+1)^j -1), {j, n}]/(k^2+k-1)^n];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 12 2021 *)
PROG
(Sage)
def A156881(n, k): return factorial(n) if (k==0) else product((k^j*(k+1)^j -1) for j in (1..n))/(k^2 +k-1)^n
flatten([[A156881(k, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 12 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 17 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 12 2021
STATUS
approved