A182969 G.f. satisfies: A(x) = 1 + x*A(x)^3*A(x*A(x)).

1, 1, 4, 23, 159, 1236, 10454, 94401, 899286, 8964253, 92961432, 998600238, 11075132605, 126489183013, 1484601117235, 17876874457054, 220546820252773, 2784446513061287, 35940592329823310, 473893641259375150

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(n+1)]*A(x)^(n+1)/(n+1)! ).

a(n)=T(n-1,1), T(n,m)=m/n*sum(k=1..n-m, sum(i=k..n-m, T(n-m,i)*k/i*binomial(2*i-k-1,i-1))*binomial(n+k-1,n-1)), n>m, T(n,n)=1. [Vladimir Kruchinin, May 07 2012]

T(n,m) = m * sum(k=1..m-m, (T(n-m,k)*binomial(n+2*k-1,n+k-1))/(n+k)) for n>m, and T(n,n) = 1. [Vladimir Kruchinin, Aug 08 2012]

Example

G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 159*x^4 + 1236*x^5 +...
Related expansions:
A(x*A(x)) = 1 + x + 5*x^2 + 35*x^3 + 287*x^4 + 2592*x^5 + 25050*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 94*x^3 + 675*x^4 + 5331*x^5 + 45274*x^6 +...
Logarithmic series:
log(A(x)) = x*A(x)^2 + [d/dx x^3*A(x)^2]*A(x)^2/2! + [d^2/dx^2 x^5*A(x)^3]*A(x)^3/3! + [d^3/dx^3 x^7*A(x)^4]*A(x)^4/4! +...

Prog

(PARI) /* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: */
{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n,
A=exp(sum(m=0, n, Dx(m, x^(2*m+1)*A^(m+1))*A^(m+1)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}
(Maxima) T(n, m):=if n=m then 1 else m/n*sum(sum(T(n-m, i)*k/i*binomial(2*i-k-1, i-1), i, k, n-m)*binomial(n+k-1, n-1), k, 1, n-m); makelist(T(n, 1), n, 1, 10); [Vladimir Kruchinin, May 07 2012]

Crossrefs

Cf. A139702, A143426, A087949, A143435.

Sequence in context: A263843 A326350 A198916 * A263186 A245110 A342988

Adjacent sequences: A182966 A182967 A182968 * A182970 A182971 A182972

Keyword

nonn

Author

Paul D. Hanna, Dec 18 2010

Status

approved