A274379 G.f. satisfies A(x) = (1 + x*A(x))^3 * (1 + x^2*A(x)^3).

1, 3, 13, 70, 429, 2842, 19794, 142758, 1056655, 7980280, 61251261, 476387379, 3746317414, 29738316330, 237968639936, 1917578268288, 15546796822656, 126728260011920, 1037987924978125, 8538459191677170, 70509828893263474, 584310452973463242, 4857624566855734836, 40501472981905806550, 338594135314564168494, 2837641019938074131463, 23835438376045780734390, 200633658871150345742269, 1692132786239339256115050, 14297391426538004065333910, 121009206594941545408186768

Offset: 0

Author

Paul D. Hanna, Aug 04 2016

Keywords

• nonn

Examples

			 G.f.: A(x) = 1 + 3*x + 13*x^2 + 70*x^3 + 429*x^4 + 2842*x^5 + 19794*x^6 + 142758*x^7 + 1056655*x^8 + 7980280*x^9 + ...
such that A(x) = 1 + 3*x*A(x) + x^2*(3*A(x)^2 + A(x)^3) + x^3*(A(x)^3 + 3*A(x)^4) + 3*x^4*A(x)^5 + x^5*A(x)^6.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A181734, A274378.

Cf. A274735, A369600.

Programs

PARI

{a(n) = my(A=1); for(i=1, n, A = (1 + x*A)^3 * (1 + x^2*A^3) + x*O(x^n) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x^2*(1+x)^3)/(1+x +x^2*O(x^n) )^3 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. satisfies: A(x) = (1/x) * Series_Reversion( x*(1 - x^2*(1+x)^3) / (1+x)^3 ).

G.f. satisfies: A( x*(1 - x^2*(1+x)^3)/(1+x)^3 ) = (1+x)^3/(1 - x^2*(1+x)^3).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(3*n+3*k+3,n-2*k). - Seiichi Manyama, Jan 27 2024