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URL: https://oeis.org/A156888

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A156888
Square array T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^7 - Sum_{m=1..5} (k+1)^m )^i ) with T(n, 0) = n!, read by antidiagonals.
5
1, 1, 1, 1, 1, 2, 1, 1, 67, 6, 1, 1, 1825, 296341, 24, 1, 1, 15021, 6075061825, 86507568379, 120, 1, 1, 74221, 3388969238841, 36886153511769270625, 1666711474847102245, 720, 1, 1, 270607, 408859932813241, 11484347898092358710495061, 408508286631392559053881969770625, 2119389029714451057320373595, 5040
OFFSET
0,6
FORMULA
T(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} ( (k+1)^7 - Sum_{m=1..5} (k+1)^m )^i ) with T(n, 0) = n! (square array).
T(n, k) = ( Product_{j=1..n} (f(k)^j -1) )/(f(k) -1)^n with T(n, 0) = n! and f(k) = ((k+1)/k)*((k+1)^7 - (k+1)^6 - (k+1)^5 + 1) (square array). - G. C. Greubel, Jun 14 2021
EXAMPLE
Square array begins as:
1, 1, 1, ...;
1, 1, 1, ...;
2, 67, 1825, ...;
6, 296341, 6075061825, ...;
24, 86507568379, 36886153511769270625, ...;
Antidiagonal triangle begins as:
1;
1, 1;
1, 1, 2;
1, 1, 67, 6;
1, 1, 1825, 296341, 24;
1, 1, 15021, 6075061825, 86507568379, 120;
1, 1, 74221, 3388969238841, 36886153511769270625, 1666711474847102245, 720; ...
MATHEMATICA
(* First program *)
T[n_, m_] = If[m==0, n!, Product[Sum[((m+1)^7 - Sum[(m+1)^r, {r, 1, 5}])^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 *)
(* Alternative: *)
f[n_]:= ((n+1)/n)*((n+1)^7 - (n+1)^6 - (n+1)^5 + 1);
T[n_, k_]= If[k==0, n!, Product[(f[k]^j -1), {j, n}]/(f[k]-1)^n];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 14 2021 *)
PROG
(SageMath)
def f(n): return ((n+1)/n)*((n+1)^7 - (n+1)^6 - (n+1)^5 + 1)
def A156888(n, k): return factorial(n) if (k==0) else product( (f(k)^j - 1) for j in (1..n))/(f(k)-1)^n
flatten([[A156888(k, n-k) for k in (0..n)] for n in (0..6)]) # 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