A156538 - OEIS

%I #8 Jan 03 2022 07:38:25

%S 2,2,2,2,-10,2,2,-31,-31,2,2,989,-406,989,2,2,81578,-16213,-16213,

%T 81578,2,2,-19816168,3777869,670556,3777869,-19816168,2,2,

%U -14445938413,2685823244,251846999,251846999,2685823244,-14445938413,2

%N Triangle T(n, k, q) = e(n, k, q) + e(n, n-k+1, q), where e(n, k, q) = ((1 - (-q)^k)/(1 + q))*e(n-1, k, q) + (-q)^(k-1)*e(n-1, k-1, q), e(n, 0, q) = e(n, n, q) = 1, and q = 3, read by rows.

%H G. C. Greubel, <a href="/A156538/b156538.txt">Rows n = 1..50 of the triangle, flattened</a>

%H R. Parthasarathy, <a href="http://arxiv.org/abs/quant-ph/0403216">q-Fermionic Numbers and Their Roles in Some Physical Problems</a>, arXiv:quant-ph/0403216, 2004.

%F T(n, k, q) = e(n, k, q) + e(n, n-k+1, q), where e(n, k, q) = ((1 - (-q)^k)/(1 + q))*e(n-1, k, q) + (-q)^(k-1)*e(n-1, k-1, q), e(n, 0, q) = e(n, n, q) = 1, and q = 3.

%e Triangle begins as:

%e   2;

%e   2,            2;

%e   2,          -10,          2;

%e   2,          -31,        -31,         2;

%e   2,          989,       -406,       989,         2;

%e   2,        81578,     -16213,    -16213,     81578,          2;

%e   2,    -19816168,    3777869,    670556,   3777869,  -19816168,            2;

%e   2, -14445938413, 2685823244, 251846999, 251846999, 2685823244, -14445938413, 2;

%t e[n_, k_, q_]:= e[n,k,q]= If[k<0 || k>n, 0, If[k==1 || k==n, 1, ((1-(-q)^k)/(1+q))*e[n-1, k, q] + (-q)^(k-1)*e[n-1, k-1, q] ]];

%t T[n_, k_, q_]:= e[n,k,q] + e[n,n-k+1,q];

%t Table[T[n, k, 3], {n, 12}, {k, n}]//Flatten (* modified by _G. C. Greubel_, Jan 03 2022 *)

%o (Sage)

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

%o     if (k<0 or k>n): return 0

%o     elif (k==1 or k==n): return 1

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

%o def A156538(n,k,q): return e(n,k,q) + e(n,n-k+1,q)

%o flatten([[A156538(n,k,3) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Jan 03 2022

%Y Cf. A156535, A156539.

%K sign,tabl

%O 1,1

%A _Roger L. Bagula_, Feb 09 2009

%E Edited by _G. C. Greubel_, Jan 03 2022