A244278 G.f. satisfies: A(x*A(-x)) = C(x), where C(x) = 1 + x*C(x)^2 is the Catalan function of A000108.

1, 1, 3, 8, 25, 87, 323, 1213, 4616, 18122, 73034, 296118, 1201985, 4944712, 20699593, 87039567, 364582540, 1536503788, 6577636822, 28301509234, 120717294622, 515067346615, 2238588931727, 9820016463003, 42372035446400, 181068476007679, 795531553724263, 3577424216585403

Offset

0,3

Links

Table of n, a(n) for n=0..27.

Formula

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

Example

G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 25*x^4 + 87*x^5 + 323*x^6 + 1213*x^7 +...
where
A(x*A(-x)) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 1430*x^8 +...+ A000108(n)*x^n +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 22*x^3 + 75*x^4 + 272*x^5 + 1034*x^6 +...
Series_Reversion(x*A(-x)) = x + x^2 - x^3 - 2*x^4 + x^5 + 10*x^6 - x^7 - 80*x^8 - 35*x^9 + 841*x^10 + 956*x^11 +...

Prog

(PARI) /* From formula: A(x*A(-x)) = C(x) = 1 + x*C(x)^2 */
{a(n)=local(A=1+x, C=sum(m=0, n, binomial(2*m, m)/(m+1)*x^m));
for(i=1, n, A=subst(C, x, serreverse(x*subst(A, x, -x) +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* A(x) = 1 + A(x)^2*Series_Reversion(x*A(-x)): */
{a(n)=local(A=1+x); for(i=1, n, A=1+A^2*serreverse(x*subst(A, x, -x) +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A000108.

Sequence in context: A148798 A148799 A148800 * A373175 A190343 A185043

Adjacent sequences: A244275 A244276 A244277 * A244279 A244280 A244281

Keyword

nonn

Author

Paul D. Hanna, Jun 24 2014

Status

approved