A156951 Triangle T(n, k, q) = t(n,q)/(t(k,q)*t(n-k,q)), where t(n, k) = Product_{j=1..n} q-Pochhammer(j, k+1, k+1)/(1-(k+1))^j and t(n, 0) = n!, with q = 2, read by rows.

1, 1, 1, 1, 4, 1, 1, 52, 52, 1, 1, 2080, 27040, 2080, 1, 1, 251680, 130873600, 130873600, 251680, 1, 1, 91611520, 5764196838400, 230567873536000, 5764196838400, 91611520, 1, 1, 100131391360, 2293297240551116800, 11099558644267405312000, 11099558644267405312000, 2293297240551116800, 100131391360, 1

Offset

0,5

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, q) = t(n,q)/(t(k,q)*t(n-k,q)), where t(n, k) = Product_{j=1..n} q-Pochhammer(j, k+1, k+1)/(1-(k+1))^j and t(n, 0) = n!, with q = 2.

Example

Triangle begins as:
  1;
  1,        1;
  1,        4,             1;
  1,       52,            52,               1;
  1,     2080,         27040,            2080,             1;
  1,   251680,     130873600,       130873600,        251680,        1;
  1, 91611520, 5764196838400, 230567873536000, 5764196838400, 91611520, 1;

Mathematica

t[n_, k_]= If[k==0, n!, Product[QPochhammer[k+1, k+1, j]/(-k)^j, {j, n}]];
T[n_, k_, q_]= t[n, q]/(t[k, q]*t[n-k, q]);
Table[T[n, k, 2], {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==0) else product( q_pochhammer(j, k+1, k+1)/(-k)^j for j in (1..n) )
def T(n, k, q): return t(n, q)/(t(k, q)*t(n-k, q))
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 08 2022

Crossrefs

Cf. A007318 (q=0), A156950 (q=1), this sequence (q=2), A156952 (q=3).

Cf. A156953.

Sequence in context: A176419 A299471 A102602 * A357052 A121066 A343635

Adjacent sequences: A156948 A156949 A156950 * A156952 A156953 A156954

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 19 2009

Extensions

Edited by G. C. Greubel, Jan 08 2022

Status

approved