A101478 G.f. satisfies A(x) = x*(1+A)^4/(1+A^2).

0, 1, 4, 21, 124, 782, 5144, 34845, 241196, 1697498, 12104872, 87246770, 634425752, 4647805372, 34267130928, 254035385949, 1892315106252, 14155536314786, 106288436980488, 800753707211430, 6050872882024520

Offset

0,3

Links

Table of n, a(n) for n=0..20.

M. Bousquet-Mélou, Limit laws for embedded trees

Formula

G.f. (1-(1-8*x)^(1/4))/(1+(1-8*x)^(1/4))-1, a(n)=sum(m=1..n, m*sum(k=0..n-m(-1)^(n-m-k)*binomial(n+k-1,n-1)*sum(j=0..k, binomial(j,n-m-3*k+2*j)*binomial(k,j)*2^(2*n-2*m-5*k+3*j)*3^(-n+m+3*k-j))))/n, n>0, a(0)=0. - Vladimir Kruchinin, Dec 10 2011

a(n) ~ 2^(3*n-1)/(Gamma(3/4)*n^(5/4)) * (1 - 2*Gamma(3/4)/ (n^(1/4)*sqrt(Pi)) + 3*Gamma(3/4)^2/(sqrt(2*n)*Pi)). - Vaclav Kotesovec, Sep 16 2013

Conjecture: n*(n-1)*(n+1)*a(n) -12*n*(n-1)*(2*n-3)*a(n-1) +12*(n-1)*(16*n^2-64*n+65)*a(n-2) -16*(2*n-5)*(4*n-9)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Nov 10 2013

Maple

A:= proc(n) option remember; if n=0 then 0 else convert(series(x* (1+A(n-1))^4/ (1+A(n-1)^2), x, n+1), polynom) fi end: a:= n-> coeff(A(n), x, n): seq(a(n), n=0..20); # Alois P. Heinz, Aug 23 2008

Mathematica

a[0]=0; a[n_] := Sum[m*Sum[(-1)^(n-m-k)*Binomial[n+k-1, n-1]*Sum[Binomial[j, n-m-3*k+2*j]*Binomial[k, j]*2^(2*n-2*m-5*k+3*j)*3^(-n+m+3*k-j), {j, 0, k}], {k, 0, n-m}], {m, 1, n}]/n; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 30 2015, after Vladimir Kruchinin *)

Prog

(Maxima)
a(n):=sum(m*sum((-1)^(n-m-k)*binomial(n+k-1, n-1)*sum(binomial(j, n-m-3*k+2*j)*binomial(k, j)*2^(2*n-2*m-5*k+3*j)*3^(-n+m+3*k-j), j, 0, k), k, 0, n-m), m, 1, n)/n; /* Vladimir Kruchinin, Dec 10 2011 */

Crossrefs

Sequence in context: A108404 A115136 A364866 * A153291 A244062 A093965

Adjacent sequences: A101475 A101476 A101477 * A101479 A101480 A101481

Keyword

nonn

Author

Ralf Stephan, Jan 21 2005

Status

approved