%I #8 Jan 03 2022 07:37:15
%S 1,1,1,1,-13,1,1,-127,395,1,1,2635,8857,-50645,1,1,113369,-1090125,
%T -6392903,25929899,1,1,-9636493,-157388911,2738123923,11163788345,
%U -53104434517,1,1,-1647840047,58603503067,1708972394545,-20846248885229,-99301333604807,435031527557803,1
%N Triangle T(n, k, q) = e(n, k, q), where e(n, k, q) = ((1 - (-q)^(n+k-1))/(1 + q))*e(n-1, k, q) + (-q)^(n+k-2)*e(n-1, k-1, q), e(n, 0, q) = e(n, n, q) = 1, and q = 2, read by rows.
%H G. C. Greubel, <a href="/A156539/b156539.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), where e(n, k, q) = ((1 - (-q)^(n+k-1))/(1 + q))*e(n-1, k, q) + (-q)^(n+k-2)*e(n-1, k-1, q), e(n, 0, q) = e(n, n, q) = 1, and q = 2.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -13, 1;
%e 1, -127, 395, 1;
%e 1, 2635, 8857, -50645, 1;
%e 1, 113369, -1090125, -6392903, 25929899, 1;
%e 1, -9636493, -157388911, 2738123923, 11163788345, -53104434517, 1;
%t e[n_, k_, q_]:= e[n,k,q]= If[k<0 || k>n, 0, If[k==1 || k==n, 1, ((1-(-q)^(n+k-1))/(1+q))*e[n-1, k, q] + (-q)^(n+k-2)*e[n-1, k-1, q] ]];
%t T[n_, k_, q_]:= e[n, k, q];
%t Table[T[n, k, 2], {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)^(n+k-1))/(1+q))*e(n-1, k, q) + (-q)^(n+k-2)*e(n-1, k-1, q)
%o def T(n,k,q): return e(n,k,q)
%o flatten([[T(n,k,2) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Jan 03 2022
%Y Cf. A156535, A156538.
%K sign,tabl
%O 1,5
%A _Roger L. Bagula_, Feb 09 2009
%E Edited by _G. C. Greubel_, Jan 03 2022
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