A206304 E.g.f. is series reversion of x - log(1+x)^2.

1, 2, 6, 22, 80, 128, -2940, -63072, -932088, -11628648, -114829968, -417677856, 21173151792, 869103962400, 23125766258208, 492858262277472, 7960636847682816, 46152793911618432, -3484964629275707328, -212667378331722666240

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..250

Formula

a(n) = (n-1)!*Sum_{k=0..n-1} binomial(n+k-1,n-1) * ( Sum_{j=0..k} binomial(k,j) * (Sum_{i=0..j} (-1)^i*binomial(j,i)*(2*(j-i))!*stirling1(n+j-i-1, 2*(j-i))/(n+j-i-1)!) )), n>0.

Lim sup n->infinity (|a(n)|/n!)^(1/n) = 1/abs((1+LambertW(-1/2))^2) = 1.57356815308645229... - Vaclav Kotesovec, Jan 22 2014

Mathematica

Rest[CoefficientList[InverseSeries[Series[x-Log[1+x]^2, {x, 0, 20}], x], x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 22 2014 *)

Prog

(Maxima) a(n):= ((n-1)!*sum(binomial(n+k-1, n-1)*sum(binomial(k, j) *sum(((-1)^(i)*binomial(j, i)*(2*(j-i))!*stirling1(n+j-i-1, 2*(j-i)))/(n+j-i-1)!, i, 0, j), j, 0, k), k, 0, n-1));
(SageMath)
m=40 # a = A206304
def b_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( x - log(1+x)^2 ).reverse().list()
a= b_list(m+1)
[factorial(n)*a[n] for n in range(1, m+1)] # G. C. Greubel, Dec 21 2022

Crossrefs

Cf. A185151.

Sequence in context: A106434 A150228 A203038 * A201372 A072547 A150229

Adjacent sequences: A206301 A206302 A206303 * A206305 A206306 A206307

Keyword

sign

Author

Vladimir Kruchinin, Feb 06 2012

Status

approved