A185157 G.f. A(x) = sum(n>0, a(n)*x^n/(2*n-1)!) is the inverse function to x*Bernoulli(x).

1, 3, 50, 2100, 166824, 21538440, 4115105280, 1091804313600, 384202115256960, 173201547619900800, 97349279409046828800, 66747386996603337024000, 54838533307770850530816000, 53185913922332495626882560000

Offset: 1

Author

Vladimir Kruchinin, Jan 23 2012

Keywords

• nonn

Comments

r(n)=sum(A191578(n,k)*k!/(n!*(n-k)!)*a(k)/(2*k-1)!,k,1,n)=0, n>1. r(1)=1.

The central column of the Worpitzky triangle, a(n) = A028246(2n, n). Peter Luschny, Jul 17 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 1..215
• Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

Crossrefs

Cf. A028246, A191578.

Programs

Mathematica

a[n_] := (n-1)!*StirlingS2[2*n-1, n]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Feb 21 2013, from 1st formula *)

Maxima

a(n)=(n-1)!*stirling2(2*n-1,n);

Sage

def A185157(n) :
    return (1/n)*add((-1)^(n-i)*binomial(n,i)*i^(2*n-1) for i in (0..n))
[A185157(n) for n in (1..14)] # Peter Luschny, Jul 17 2012

Formula

a(n) = (n-1)!*stirling2(2*n-1,n).

a(n) = (1/n)*sum{i=0..n}(-1)^(n-i)*binomial(n,i)*i^(2*n-1) - Peter Luschny, Jul 17 2012

O.g.f.: Sum_{n>=1} n^(2*n-2)*x^n/(1 + n^2*x)^n = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Jan 06 2018