A240996 G.f. satisfies: A(x)^2 = x + A(x*A(x)^2).

1, 1, 1, 5, 41, 470, 6804, 118365, 2398095, 55393202, 1436315357, 41309995331, 1305311240677, 44956819853455, 1676510128660807, 67307814275738181, 2894812673176510587, 132795587656049202117, 6472720746082336622865, 334076240871194943910092

Offset

0,4

Comments

Self-convolution yields A088223.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

G.f. A(x) satisfies:

(1) A(x) = B(x)^2 - x/B(x)^2 where B(x) = A(x/B(x)^2) = sqrt(x/Series_Reversion(x*A(x)^2)).

(2) A( x*A(x)^6 - 2*x^2*A(x)^4 + x^3*A(x)^2 ) = A(x)^4 - 3*x*A(x)^2 + x^2.

a(n) ~ c * 2^n * n^(n - 1/2 - log(2)/4) / (exp(n) * (log(2))^n), where c = 0.411579248322849751402... . - Vaclav Kotesovec, Aug 08 2014

Example

G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 41*x^4 + 470*x^5 + 6804*x^6 +...
Compare these related series:
A(x)^2 = 1 + 2*x + 3*x^2 + 12*x^3 + 93*x^4 + 1032*x^5 + 14655*x^6 +...
A(x*A(x)^2) = 1 + x + 3*x^2 + 12*x^3 + 93*x^4 + 1032*x^5 + 14655*x^6 +...

Prog

(PARI) {a(n)=local(A=[1, 1], Ax); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A]=Vec(1+subst(Ax, x, x*Ax^2) - Ax^2)[#A]); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A088223, A240999, A241996, A241997, A241998, A241999.

Sequence in context: A332236 A305981 A032188 * A346982 A143415 A056545

Adjacent sequences: A240993 A240994 A240995 * A240997 A240998 A240999

Keyword

nonn

Author

Paul D. Hanna, Aug 06 2014

Status

approved