A154677 G.f. satisfies: A(x/A(x)) = C(x) where C(x) = 1 + x*C(x)^2 is the g.f. of A000108 (Catalan numbers).

1, 1, 3, 13, 70, 440, 3116, 24274, 204407, 1836339, 17425275, 173329307, 1796783304, 19323703019, 214843877103, 2462522274426, 29032815570544, 351447240945518, 4361579736404011, 55424256247911490, 720399315622779670, 9569215299494074698, 129799982362958621827

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

G.f. satisfies: A( (x-x^2) / A(x-x^2) ) = 1/(1-x).

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

G.f. satisfies: A(x) = 1 + A(x)^2*Series_Reversion(x/A(x)). - Paul D. Hanna, Dec 06 2009

Example

G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 70*x^4 + 440*x^5 + ... where
A(x/A(x)) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ... + A000108(n)*x^n + ...
x/A(x) = x - x^2 - 2*x^3 - 8*x^4 - 43*x^5 - 277*x^6 - 2026*x^7 - ...

Prog

(PARI) {a(n) = my(A=1+x, C = sum(k=0, n, binomial(2*k+1, k)/(2*k+1)*x^k) +x*O(x^n)); for(k=2, n, A = truncate(A); A = subst(C, x, serreverse(x/(A + x^2*O(x^k))))); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)} \\ program revised by Paul D. Hanna, Mar 30 2026
upto(30)
(PARI) {a(n) = my(A=1+x); for(k=2, n, A = truncate(A); A = 1 + A^2*serreverse(x/(A + O(x^k)))); polcoef(GF=A, n)} \\ Paul D. Hanna, Dec 06 2009
{upto(n) = a(n); Vec(GF)} \\ program revised by Paul D. Hanna, Mar 30 2026
upto(30)

Crossrefs

Cf. A000108.

Cf. variants: A168448, A168478. - Paul D. Hanna, Dec 06 2009

Sequence in context: A059726 A274379 A192209 * A121586 A024337 A001495

Adjacent sequences: A154674 A154675 A154676 * A154678 A154679 A154680

Keyword

nonn

Author

Paul D. Hanna, Jan 14 2009

Status

approved