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%I #7 Jun 19 2018 22:13:05
%S 1,-1,1,5,-11,-91,-419,-1555,35029,708629,8413261,79666685,-294564731,
%T -38505298651,-1052947792259,-18923930396275,-206463542201291,
%U 1794180062198069,205343758433071021,8230374933815425565,237203632846737093349,4859533645922850398789,34618271271121471451101
%N Expansion of e.g.f. 1/(1 + (exp(x) - 1)/(1 + (exp(x) - 1)^2/(1 + (exp(x) - 1)^3/(1 + ...)))), a continued fraction.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction</a>
%F a(n) = Sum_{k=0..n} Stirling2(n,k)*A007325(k)*k!.
%e E.g.f.: A(x) = 1 - x + x^2/2! + 5*x^3/3! - 11*x^4/4! - 91*x^5/5! - 419*x^6/6! - 1555*x^7/7! + ...
%t nmax = 22; CoefficientList[Series[1/(1 + ContinuedFractionK[(Exp[x] - 1)^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
%t b[n_] := b[n] = SeriesCoefficient[QPochhammer[x, x^5] QPochhammer[x^4, x^5]/(QPochhammer[x^2, x^5] QPochhammer[x^3, x^5]), {x, 0, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k] k!, {k, 0, n}]; Table[a[n], {n, 0, 22}]
%Y Cf. A007325, A019538, A301921.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, Jun 19 2018
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