A173139 G.f. satisfies: A(x) = x*A(x) + A(x^2*A(x)^2).

1, 1, 2, 4, 11, 31, 97, 309, 1026, 3466, 11964, 41856, 148373, 531233, 1919313, 6986745, 25604367, 94379887, 349702606, 1301729084, 4865680876, 18255350676, 68724316775, 259521249065, 982796892300, 3731493081316, 14201727734640

Offset

0,3

Links

Table of n, a(n) for n=0..26.

Formula

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

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

Given e.g.f. E(x), then E(x)/exp(x) is an even function.

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 31*x^5 + 97*x^6 +...
Let G(x) = x + A(x^2):
G(x) = 1 + x + x^2 + 2*x^4 + 4*x^6 + 11*x^8 + 31*x^10 + 97*x^12 +...
then A(x) = G(x*A(x)) and G(x) = A(x/G(x)).
Given e.g.f. E(x), then E(x)/exp(x) is the even function:
E(x)/exp(x) = 1 + x^2/2! + 4*x^4/4! + 21*x^6/6! + 129*x^8/8! + 863*x^10/10! + 6109*x^12/12! +...

Prog

(PARI) {a(n)=local(A=1+x+x^2+x*O(x^n), B); for(i=1, #binary(n)+1, A=x+subst(A, x, x^2+x*O(x^n)); A=(1/x)*serreverse(x/A)); polcoeff(A, n)}

Crossrefs

Sequence in context: A118974 A119020 A073191 * A148164 A364594 A213091

Adjacent sequences: A173136 A173137 A173138 * A173140 A173141 A173142

Keyword

nonn

Author

Paul D. Hanna, Feb 10 2010

Status

approved