A143426 G.f. satisfies: A(x) = 1 + x*A(x*A(x))^2.

1, 1, 2, 7, 32, 175, 1086, 7429, 54994, 435120, 3647686, 32192596, 297654824, 2872372828, 28841766844, 300592170551, 3244942353856, 36219458512421, 417365572999944, 4958429472475171, 60659660219655616, 763325035692109389

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..235

Formula

G.f. satisfies: x - G(x) = G(x)^2*A(x)^2 where G(x*A(x)) = x.

G.f. satisfies: A(x) = exp( Sum_{n>=0} [d^n/dx^n x^(2n+1)*A(x)^(2n+2)]*A(x)^(-2n-2)/(n+1)! ). [Paul D. Hanna, Dec 18 2010]

Example

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 175*x^5 + 1086*x^6 +...
A(x*A(x)) = 1 + x + 3*x^2 + 13*x^3 + 70*x^4 + 434*x^5 + 2986*x^6 +...
A(x*A(x))^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 175*x^4 + 1086*x^5 +...
Logarithmic series:
log(A(x)) = x + [d/dx x^3*A(x)^4]*A(x)^(-4)/2! + [d^2/dx^2 x^5*A(x)^6]*A(x)^(-6)/3! + [d^3/dx^3 x^7*A(x)^8]*A(x)^(-8)/4! +...

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+x*subst(A^2, x, x*A)); polcoeff(A, n)}
(PARI) /* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, Dec 18 2010] */
{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^(2*m+2))*A^(-2*m-2)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}

Crossrefs

Cf. A139702, A087949, A143435, A182969.

Sequence in context: A191809 A161392 A161393 * A125223 A339226 A277359

Adjacent sequences: A143423 A143424 A143425 * A143427 A143428 A143429

Keyword

nonn

Author

Paul D. Hanna, Aug 14 2008

Status

approved