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)^4 - (k+1)^3)^i ) with T(n, 0) = n! (square array).
T(n, k) = ( Product_{j=1..n} (k^j*(k+1)^(3*j) -1) )/(k*(k+1)^3 -1)^n with T(n, 0) = n! (square array).
EXAMPLE
Square array begins as:
1, 1, 1, 1, ...;
1, 1, 1, 1, ...;
2, 9, 55, 193, ...;
6, 657, 163405, 7152001, ...;
24, 384345, 26215881175, 50886093754945, ...;
Triangle begins as:
1;
1, 1;
1, 1, 2;
1, 1, 9, 6;
1, 1, 55, 657, 24;
1, 1, 193, 163405, 384345, 120;
1, 1, 501, 7152001, 26215881175, 1799118945, 720;
1, 1, 1081, 125501001, 50886093754945, 227121050616681925, 67375205371305, 5040;
MATHEMATICA
(* First program *)
T[n_, m_] = If[m==0, n!, Product[Sum[(-(m+1)^3 + (m+1)^4)^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)^(3*j) -1), {j, n}]/(k*(k+1)^3 -1)^n];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 14 2021 *)
PROG
(Sage)
def A156883(n, k): return factorial(n) if (k==0) else product((k^j*(k+1)^(3*j) -1) for j in (1..n))/(k*(k+1)^3 -1)^n
flatten([[A156883(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