A156952 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 = 3, read by rows.

1, 1, 1, 1, 5, 1, 1, 105, 105, 1, 1, 8925, 187425, 8925, 1, 1, 3043425, 5432513625, 5432513625, 3043425, 1, 1, 4154275125, 2528644954460625, 214934821129153125, 2528644954460625, 4154275125, 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 = 3.

Example

Triangle begins as:
  1;
  1,       1;
  1,       5,          1;
  1,     105,        105,          1;
  1,    8925,     187425,       8925,       1;
  1, 3043425, 5432513625, 5432513625, 3043425, 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, 3], {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, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 08 2022

Crossrefs

Cf. A007318 (q=0), A156950 (q=1), A156951 (q=2), this sequence (q=3).

Cf. A156953.

Sequence in context: A106238 A173475 A174919 * A158748 A351241 A273874

Adjacent sequences: A156949 A156950 A156951 * A156953 A156954 A156955

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 19 2009

Extensions

Edited by G. C. Greubel, Jan 08 2022

Status

approved