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A156950
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 = 1, read by rows.
4
1, 1, 1, 1, 3, 1, 1, 21, 21, 1, 1, 315, 2205, 315, 1, 1, 9765, 1025325, 1025325, 9765, 1, 1, 615195, 2002459725, 30036895875, 2002459725, 615195, 1, 1, 78129765, 16021680259725, 7450081320772125, 7450081320772125, 16021680259725, 78129765, 1
OFFSET
0,5
REFERENCES
Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 182.
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 = 1.
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 3, 1;
1, 21, 21, 1;
1, 315, 2205, 315, 1;
1, 9765, 1025325, 1025325, 9765, 1;
1, 615195, 2002459725, 30036895875, 2002459725, 615195, 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, 1], {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, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 08 2022
CROSSREFS
Cf. A007318 (q=0), this sequence (q=1), A156951 (q=2), A156952 (q=3).
Cf. A156953.
Sequence in context: A111382 A173884 A176418 * A083998 A277170 A216922
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 19 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 08 2022
STATUS
approved