A156220 - OEIS

%I #10 Jan 02 2022 07:26:18

%S -2,-2,3,-2,3,-1,-2,3,-1,109,-2,3,-1,325,1555523,-2,3,-1,973,32671835,

%T 49621794478165,-2,3,-1,2917,686126051,15630874866123949,

%U 27744919164118690798376051,-2,3,-1,8749,14408699579,4923725784550050421,270929135785330782929292449579,2134369240927848351630724472718209102550421

%N Triangle T(n, k) = (2^k/3)*Q(k, n), with T(0, 0) = -2, where Q(k, n) = (1/2)*( -Q(k-1, n) + 3*p(2, k-1)^n), and p(q, n) = Product_{j=1..n} ( (1-x^k)/(1-x) ), read by rows.

%C A triangle sequence based on Carlitz q-Eulerian formulas (see ref).

%H G. C. Greubel, <a href="/A156220/b156220.txt">Rows n = 0..25 of the triangle, flattened</a>

%H L. Carlitz, <a href="https://projecteuclid.org/journals/duke-mathematical-journal/volume-15/issue-4/q-Bernoulli-numbers-and-polynomials/10.1215/S0012-7094-48-01588-9.short">q-Bernoulli numbers and polynomials</a>, Duke Math. J. Volume 15, Number 4 (1948), 987-1000.

%F T(n, k) = (2^k/3)*Q(k, n), with T(0, 0) = -2, where Q(k, n) = (1/2)*( -Q(k-1, n) + 3*p(2, k-1)^n), and p(q, n) = Product_{j=1..n} ( (1-q^k)/(1-q) ).

%e Triangle begins as:

%e   -2;

%e   -2, 3;

%e   -2, 3, -1;

%e   -2, 3, -1,  109;

%e   -2, 3, -1,  325,   1555523;

%e   -2, 3, -1,  973,  32671835,    49621794478165;

%e   -2, 3, -1, 2917, 686126051, 15630874866123949, 27744919164118690798376051;

%t Q[x_, n_]:= Q[x, n]= If[n==0, 1, If[x==0, -6, (1/2)*(-Q[x-1, n] + 3*((-1)^(k-1)*QPochhammer[2, 2, x-1])^n)]];

%t T[n_, k_]:= If[n==0, -2, (2^k/3)*Q[k, n]];

%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Dec 31 2021 *)

%o (Sage)

%o from sage.combinat.q_analogues import q_pochhammer

%o @CachedFunction

%o def Q(k,n):

%o     if (n==0): return 1

%o     elif (k==0): return -6

%o     else: return (1/2)*( -Q(k-1,n) + 3*(-1)^(n*(k-1))*(q_pochhammer(k-1,2,2))^n)

%o def T(n,k): return -2 if (n==0) else (2^k/3)*Q(k,n)

%o flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Dec 31 2021

%Y Cf. A156222.

%K sign,tabl

%O 0,1

%A _Roger L. Bagula_, Feb 06 2009

%E Edited by _G. C. Greubel_, Dec 31 2021