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A301921
Expansion of e.g.f. 1/(1 - (exp(x) - 1)/(1 - (exp(x) - 1)^2/(1 - (exp(x) - 1)^3/(1 - ...)))), a continued fraction.
3
1, 1, 3, 19, 159, 1651, 21303, 324619, 5653119, 110909251, 2424648903, 58430418619, 1537673312079, 43860906193651, 1347852526593303, 44392923532503019, 1560023977386027039, 58259266750803410851, 2303999137417453606503, 96188099015599819297819, 4227325636692027926037999
OFFSET
0,3
LINKS
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction
FORMULA
a(n) = Sum_{k=0..n} Stirling2(n,k)*A005169(k)*k!.
a(n) ~ c * d^n * n!, where d = 2.19787763261059933075080498218168228... and c = 0.250957960982243982921501085974065... - Vaclav Kotesovec, Dec 20 2018
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 159*x^4/4! + 1651*x^5/5! + 21303*x^6/6! + ...
MATHEMATICA
nmax = 20; 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[1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, n}]), {x, 0, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k] k!, {k, 0, n}]; Table[a[n], {n, 0, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jun 19 2018
STATUS
approved