A156222 - OEIS

%I #12 Jan 03 2022 11:13:13

%S -2,-6,9,-18,21,-15,-54,57,-51,375,-162,165,-159,1131,4666413,-486,

%T 489,-483,3399,98015025,148865383434975,-1458,1461,-1455,10203,

%U 2058376701,46892624598373299,83234757492356072395126701

%N 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.

%H G. C. Greubel, <a href="/A156222/b156222.txt">Rows n = 0..19 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), pp. 987 - 1000.

%F 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.

%e Triangle begins as:

%e     -2;

%e     -6,   9;

%e    -18,  21,  -15;

%e    -54,  57,  -51,  375;

%e   -162, 165, -159, 1131,  4666413;

%e   -486, 489, -483, 3399, 98015025, 148865383434975;

%t 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 T[n_, k_, q_]:= If[n==0, -2, 2^k*Q[k, n, q]];

%t Table[T[n, k, 2], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 01 2022 *)

%o (Sage)

%o from sage.combinat.q_analogues import q_pochhammer

%o @CachedFunction

%o def Q(k,n,q):

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

%o     elif (k==0): return -q*(q+1)^n

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

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

%o flatten([[T(n,k,2) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jan 01 2022

%Y Cf. A156220.

%K sign,tabl

%O 0,1

%A _Roger L. Bagula_, Feb 06 2009

%E Edited by _G. C. Greubel_, Jan 01 2022