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A156222 Triangle T(n, k, q) = q^k*Q(k, n, q), with T(0, 0, q) = -2, where Q(k, n, q) = (1/q)*( -Q(k-1, n, q) + (1+q)*p(q, k-1)^n), Q(k, 0, q) = -q*(1+q)^n, p(q, n) = Product_{j=1..n} ( (1-q^k)/(1-q) ), and q = 2, read by rows. 3
-2, -6, 9, -18, 21, -15, -54, 57, -51, 375, -162, 165, -159, 1131, 4666413, -486, 489, -483, 3399, 98015025, 148865383434975, -1458, 1461, -1455, 10203, 2058376701, 46892624598373299, 83234757492356072395126701 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
L. Carlitz, q-Bernoulli numbers and polynomials Duke Math. J. Volume 15, Number 4 (1948), pp. 987 - 1000.
FORMULA
T(n, k, q) = q^k*Q(k, n, q), with T(0, 0, q) = -2, where Q(k, n, q) = (1/q)*( -Q(k-1, n, q) + (1+q)*p(q, k-1)^n), Q(k, 0, q) = -q*(1+q)^n, p(q, n) = Product_{j=1..n} ( (1-q^k)/(1-q) ), and q = 2.
EXAMPLE
Triangle begins as:
-2;
-6, 9;
-18, 21, -15;
-54, 57, -51, 375;
-162, 165, -159, 1131, 4666413;
-486, 489, -483, 3399, 98015025, 148865383434975;
MATHEMATICA
Q[k_, n_, q_]:= Q[k, n, q]= If[n==0, 1, If[k==0, -q*(1+q)^n, (1/q)*( -Q[k-1, n, q] + (1+q)*(-1)^(n*(k-1))*QPochhammer[q, q, k-1]^n ) ]];
T[n_, k_, q_]:= If[n==0, -2, 2^k*Q[k, n, q]];
Table[T[n, k, 2], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 01 2022 *)
PROG
(Sage)
from sage.combinat.q_analogues import q_pochhammer
@CachedFunction
def Q(k, n, q):
if (n==0): return 1
elif (k==0): return -q*(q+1)^n
else: return (1/q)*(-Q(k-1, n, q) + (1+q)*((-1)^(k-1)*q_pochhammer(k-1, q, q))^n)
def T(n, k, q): return -2 if (n==0) else q^k*Q(k, n, q)
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jan 01 2022
CROSSREFS
Cf. A156220.
Sequence in context: A072481 A032471 A358258 * A002886 A028724 A222048
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Feb 06 2009
EXTENSIONS
Edited by G. C. Greubel, Jan 01 2022
STATUS
approved

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Last modified July 23 13:44 EDT 2024. Contains 374549 sequences. (Running on oeis4.)