A156953 Array A(n, k) = Product_{j=1..n} Product_{i=1..j} (1 - (k+1)^i)/(1 - (k+1))^j, with A(n, k) = n!, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 63, 24, 1, 1, 5, 208, 19845, 120, 1, 1, 6, 525, 432640, 193786425, 720, 1, 1, 7, 1116, 4685625, 108886835200, 119216439727875, 5040, 1, 1, 8, 2107, 32381856, 14260348265625, 9975288480661504000, 9314352420075537699375, 40320

Offset

0,6

References

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 182.

Links

G. C. Greubel, Rows n = 0..20 of the triangle, flattened

Formula

T(n, k) = A(k, n-k) where the array is given by A(n, k) = Product_{j=1..n} Product_{i=1..j} (1 - (k+1)^i)/(1 - (k+1))^j, with A(n, k) = n!.

T(n, k) = Product_{j=1..k} (q-Pochhammer(j, n-k+1, n-k+1)/(-k)^j) with T(n, n) = n!. - G. C. Greubel, Jan 08 2022

Example

  1;
  1, 1;
  1, 1, 2;
  1, 1, 3,    6;
  1, 1, 4,   63,      24;
  1, 1, 5,  208,   19845,          120;
  1, 1, 6,  525,  432640,    193786425,             720;
  1, 1, 7, 1116, 4685625, 108886835200, 119216439727875, 5040;

Mathematica

A[n_, k_]:= If[k==0, n!, Product[QPochhammer[k+1, k+1, j]/(-k)^j, {j, n}]];
T[n_, k_]:= A[k, n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 08 2022 *)

Prog

(Sage)
from sage.combinat.q_analogues import q_pochhammer
def T(n, k): return factorial(n) if (k==n) else product( q_pochhammer(j, n-k+1, n-k+1)/(k-n)^j for j in (1..k) )
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 08 2022

Crossrefs

Cf. A156950, A156951, A156952.

Sequence in context: A123353 A156540 A156582 * A293785 A156881 A376554

Adjacent sequences: A156950 A156951 A156952 * A156954 A156955 A156956

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 19 2009

Extensions

Edited by G. C. Greubel, Jan 08 2022

Status

approved