A156220 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.

-2, -2, 3, -2, 3, -1, -2, 3, -1, 109, -2, 3, -1, 325, 1555523, -2, 3, -1, 973, 32671835, 49621794478165, -2, 3, -1, 2917, 686126051, 15630874866123949, 27744919164118690798376051, -2, 3, -1, 8749, 14408699579, 4923725784550050421, 270929135785330782929292449579, 2134369240927848351630724472718209102550421

Offset: 0

Author

Roger L. Bagula, Feb 06 2009

Keywords

• sign
• tabl

Comments

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

Examples

			Triangle begins as:
  -2;
  -2, 3;
  -2, 3, -1;
  -2, 3, -1,  109;
  -2, 3, -1,  325,   1555523;
  -2, 3, -1,  973,  32671835,    49621794478165;
  -2, 3, -1, 2917, 686126051, 15630874866123949, 27744919164118690798376051;
		

Links

• G. C. Greubel, Rows n = 0..25 of the triangle, flattened
• L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. Volume 15, Number 4 (1948), 987-1000.

Crossrefs

Cf. A156222.

Programs

Mathematica

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[n_, k_]:= If[n==0, -2, (2^k/3)*Q[k, n]];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Dec 31 2021 *)

Sage

from sage.combinat.q_analogues import q_pochhammer
@CachedFunction
def Q(k,n):
    if (n==0): return 1
    elif (k==0): return -6
    else: return (1/2)*( -Q(k-1,n) + 3*(-1)^(n*(k-1))*(q_pochhammer(k-1,2,2))^n)
def T(n,k): return -2 if (n==0) else (2^k/3)*Q(k,n)
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Dec 31 2021

Formula

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) ).

Extensions

Edited by G. C. Greubel, Dec 31 2021