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

0,1

Comments

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

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.

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) ).

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A156222.

Sequence in context: A300817 A341417 A230140 * A261653 A347658 A343387

Adjacent sequences: A156217 A156218 A156219 * A156221 A156222 A156223

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 06 2009

Extensions

Edited by G. C. Greubel, Dec 31 2021

Status

approved