A228987 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^3*A(x)^8.

1, 1, 3, 13, 66, 364, 2116, 12768, 79222, 502297, 3240120, 21196593, 140295584, 937787728, 6321624862, 42926227470, 293350136170, 2015999854478, 13923926272607, 96598395025615, 672852440805930, 4703751150849738, 32991286060134402, 232091541493091566

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Formula

G.f. satisfies: A(x) = 1/A(-x*A(x)^5); note that the function G(x) = 1 + x*G(x)^3 (g.f. of A001764) also satisfies this condition: G(x) = 1/G(-x*G(x)^5).

a(n) ~ sqrt((3 - r*s^2)/(2*Pi*(3 + 28*r^2*s^5))) / (4*n^(3/2)*r^(n + 1/2)), where r = 0.1331154541373089587498695338172936885734070972340... and s = 1.408602671059676188189711196409966797670750551605... are real roots of the system of equations 1 + r*s^3 + r^3*s^8 = s, 3*r*s^2 + 8*r^3*s^7 = 1. - Vaclav Kotesovec, Nov 22 2017

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(3*n-k+1,n-2*k)/(3*n-k+1). - Seiichi Manyama, Aug 28 2023

Example

G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 66*x^4 + 364*x^5 + 2116*x^6 +...
Related expansions:
A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 312*x^4 + 1788*x^5 + 10686*x^6 +...
A(x)^8 = 1 + 8*x + 52*x^2 + 328*x^3 + 2082*x^4 + 13384*x^5 + 87124*x^6 +...
Given (1) A(x) = 1 + x*A(x)^3 + x^3*A(x)^8,
suppose (2) A(x) = 1/A(-x*A(x)^5),
then substituting x in (1) with -x*A(x)^5 yields:
1/A(x) = 1 - x*A(x)^5/A(x)^3 - x^3*A(x)^15/A(x)^8,
which illustrates that (2) is consistent with (1).

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^3+x^3*A^8 +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A216490, A112806.

Sequence in context: A156181 A260783 A373932 * A112807 A219537 A045743

Adjacent sequences: A228984 A228985 A228986 * A228988 A228989 A228990

Keyword

nonn

Author

Paul D. Hanna, Sep 10 2013

Status

approved