A156952 - OEIS

%I #6 Jan 09 2022 02:32:21

%S 1,1,1,1,5,1,1,105,105,1,1,8925,187425,8925,1,1,3043425,5432513625,

%T 5432513625,3043425,1,1,4154275125,2528644954460625,

%U 214934821129153125,2528644954460625,4154275125,1

%N 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.

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

%H G. C. Greubel, <a href="/A156952/b156952.txt">Rows n = 0..20 of the triangle, flattened</a>

%F 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.

%e Triangle begins as:

%e   1;

%e   1,       1;

%e   1,       5,          1;

%e   1,     105,        105,          1;

%e   1,    8925,     187425,       8925,       1;

%e   1, 3043425, 5432513625, 5432513625, 3043425, 1;

%t t[n_, k_]= If[k==0, n!, Product[QPochhammer[k+1, k+1, j]/(-k)^j, {j, n}]];

%t T[n_, k_, q_]= t[n, q]/(t[k, q]*t[n-k, q]);

%t Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 08 2022 *)

%o (Sage)

%o from sage.combinat.q_analogues import q_pochhammer

%o 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) )

%o def T(n,k,q): return t(n,q)/(t(k,q)*t(n-k,q))

%o flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 08 2022

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

%Y Cf. A156953.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 19 2009

%E Edited by _G. C. Greubel_, Jan 08 2022