A201795 E.g.f. satisfies: A(x)+1/2*A(x)^2 = x*exp(A(x)).

1, 1, 3, 13, 80, 621, 5887, 65689, 844587, 12289825, 199702646, 3584177829, 70418168977, 1503204079573, 34644744039375, 857391850897201, 22677415997829788, 638386960029846921, 19057447729907765407, 601346850250707128125

Offset

1,3

Links

Table of n, a(n) for n=1..20.

Formula

a(n) = n!*T(n,1), T(n,m) = Sum_{k=1..n-m} T(n-m,k)*m^k/k!-binomial(m,k)/2^k*T(n,k+m), n>m, with T(n,n)=1.

a(n) = (n-1)!*Sum_{k=1..n-1} C(n+k-1,n-1)*Sum_{j=1..k} (-1)^j*C(k,j)*Sum_{i=0..n-1} (-1)^i*j^i*C(j,n-i-1)*2^(-n+i+1)/i!, n>1, a(n)=1. - Vladimir Kruchinin, Feb 24 2012

a(n) ~ 2^(-1/4) * exp((sqrt(2)-1)*n) * (sqrt(2)-1)^(n-1/2) * n^(n-1). - Vaclav Kotesovec, Aug 04 2014

Mathematica

Rest[CoefficientList[InverseSeries[Series[(x*(2 + x))/(2*E^x), {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Aug 04 2014 *)

Prog

(Maxima)
array(B, 100, 100);
fillarray (B, makelist (-1, i, 1, 1000));
T(n, m):=if B[n, m]=-1 then BB[n, m]:(if n=m then 1 else sum(T(n-m, k)*m^k/k!-binomial(m, k)/2^k*T(n, k+m), k, 1, n-m)) else B[n, m];
makelist(n!*T(n, 1), n, 1, 20);
a(n):=if n=1 then 1 else ((n-1)!*sum(binomial(n+k-1, n-1) *sum((-1)^(j) *binomial(k, j)*sum(((-1)^i*j^i*binomial(j, n-i-1) *2^(-n+i+1))/i!, i, 0, n-1), j, 1, k), k, 1, n-1)); /* Vladimir Kruchinin, Feb 24 2012 */

Crossrefs

Sequence in context: A258377 A335636 A366179 * A308521 A183278 A331643

Adjacent sequences: A201792 A201793 A201794 * A201796 A201797 A201798

Keyword

nonn

Author

Vladimir Kruchinin, Dec 05 2011

Status

approved