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A244278 G.f. satisfies: A(x*A(-x)) = C(x), where C(x) = 1 + x*C(x)^2 is the Catalan function of A000108. 0
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 (list; graph; refs; listen; history; text; internal format)
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 * A190343 A185043 A033483

Adjacent sequences:  A244275 A244276 A244277 * A244279 A244280 A244281

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 24 2014

STATUS

approved

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Last modified May 23 19:07 EDT 2019. Contains 323528 sequences. (Running on oeis4.)