A185151 E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! is inverse function to exp(x) - x^2 - 1.

1, 1, 2, 4, -6, -232, -3116, -34652, -331680, -2206128, 9303480, 812562672, 22705836048, 484588970448, 8345456974368, 94936573618176, -635010052507872, -88746666011316480, -3781485264943422528

Offset

1,3

Links

Table of n, a(n) for n=1..19.

Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

Formula

a(n) = ((n-1)!*sum(k=1..n-1, binomial(n+k-1,n-1)*sum(j=1..k, (-1)^(j)*binomial(k,j)*sum(l=0..min(j,floor((n+j-1)/2)), (binomial(j,l)*(j-l)!*(-1)^l*Stirling2(n-2*l+j-1,j-l))/(n-2*l+j-1)!)))), n>1, a(1)=1.

Lim sup n->infinity (|a(n)|/n!)^(1/n) = 1/abs((1+LambertW(-1/2))^2) = 1.57356815308645229... - Vaclav Kotesovec, Jan 23 2014

Mathematica

Rest[CoefficientList[InverseSeries[Series[E^x-x^2-1, {x, 0, 20}], x], x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 23 2014 *)

Prog

(Maxima)  a(n):=if n=1 then 1 else ((n-1)!*sum(binomial(n+k-1, n-1)*sum((-1)^(j)*binomial(k, j)*sum((binomial(j, l)*(j-l)!*(-1)^l*stirling2(n-2*l+j-1, j-l))/(n-2*l+j-1)!, l, 0, min(j, floor((n+j-1)/2))), j, 1, k), k, 1, n-1));

Crossrefs

Cf. A206304.

Sequence in context: A345269 A345092 A033319 * A218087 A090315 A083753

Adjacent sequences: A185148 A185149 A185150 * A185152 A185153 A185154

Keyword

sign

Author

Vladimir Kruchinin, Jan 23 2012

Status

approved