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 A156535 A recursion triangle sequence:f(q,k)=(1 - (-q)^k)/(1 + q);q=2; e(n,k)= f(q, k)*e(n - 1, k) + (-q)^(k - 1)e(n - 1, k - 1). 0
 1, 1, 1, 1, -3, 1, 1, 1, -9, 1, 1, -3, -23, 67, 1, 1, 1, -81, -151, 1083, 1, 1, -3, -239, 1403, 9497, -34677, 1, 1, 1, -729, -5103, 126915, 424313, -2219285, 1, 1, -3, -2183, 31347, 1314417, -12971853, -68273223, 284068395, 1, 1, 1, -6561, -139271, 14960139 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums are: {1, 2, -1, -6, 43, 854, -24017, -1673886, 204166899, 53793882958,...}, I'm not certain I have this recursion right from the R.Parthasarathy paper. He says these are related to Stirling 2nd numbers. LINKS R. Parthasarathy, q-Fermionic Numbers and Their Roles in Some Physical Problems, arxiv:quant-ph/0403216 FORMULA f(q,k)=(1 - (-q)^k)/(1 + q);q=2; e(n,k)= f(q, k)*e(n - 1, k) + (-q)^(k - 1)e(n - 1, k - 1). EXAMPLE {1}, {1, 1}, {1, -3, 1}, {1, 1, -9, 1}, {1, -3, -23, 67, 1}, {1, 1, -81, -151, 1083, 1}, {1, -3, -239, 1403, 9497, -34677, 1}, {1, 1, -729, -5103, 126915, 424313, -2219285, 1}, {1, -3, -2183, 31347, 1314417, -12971853, -68273223, 284068395, 1}, {1, 1, -6561, -139271, 14960139, 230347569, -3765947181, -15406841031, 72721509291, 1} MATHEMATICA Clear[e, n, k, q]; f[q_, k_] := (1 - (-q)^k)/(1 + q); q = 2; e[n_, 0] := 0; e[n_, 1] := 1; e[n_, n_] := 1; e[n_, k_] := 0 /; k >= n + 1; e[n_, k_] := f[q, k]*e[n - 1, k] + (-q)^(k - 1)e[n - 1, k - 1]; Table[Table[e[n, k], {k, 1, n}], {n, 1, 10}]; Flatten[%] CROSSREFS Sequence in context: A005765 A263159 A229142 * A327564 A243748 A307847 Adjacent sequences:  A156532 A156533 A156534 * A156536 A156537 A156538 KEYWORD sign,tabl,uned AUTHOR Roger L. Bagula, Feb 09 2009 STATUS approved

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Last modified September 24 14:00 EDT 2020. Contains 337321 sequences. (Running on oeis4.)