A301923 Expansion of e.g.f. 1/(1 + (exp(x) - 1)/(1 + (exp(x) - 1)^2/(1 + (exp(x) - 1)^3/(1 + ...)))), a continued fraction.
1, -1, 1, 5, -11, -91, -419, -1555, 35029, 708629, 8413261, 79666685, -294564731, -38505298651, -1052947792259, -18923930396275, -206463542201291, 1794180062198069, 205343758433071021, 8230374933815425565, 237203632846737093349, 4859533645922850398789, 34618271271121471451101
Offset: 0
Keywords
Examples
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! + ...
Links
- N. J. A. Sloane, Transforms
- Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction
Programs
-
Mathematica
nmax = 22; CoefficientList[Series[1/(1 + ContinuedFractionK[(Exp[x] - 1)^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]! 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}]
Formula
a(n) = Sum_{k=0..n} Stirling2(n,k)*A007325(k)*k!.