A363310 Expansion of g.f. A(x) satisfying A(x) = 1 + x*G(x)^5, where G(x) = 1 + x*(G(x)^3 + G(x)^5) is the g.f. of A363311.

1, 1, 10, 120, 1620, 23560, 360352, 5714800, 93129840, 1550132320, 26242225600, 450448137216, 7821608426880, 137145465358080, 2424899712359680, 43186456105340160, 774013543036174080, 13949937641606981120, 252666943472167541760, 4596736161565468815360

Offset: 0

Author

Paul D. Hanna, May 29 2023

Keywords

• nonn

Examples

			G.f.: A(x) =  = 1 + x + 10*x^2 + 120*x^3 + 1620*x^4 + 23560*x^5 + 360352*x^6 + 5714800*x^7 + 93129840*x^8 + 1550132320*x^9 + 26242225600*x^10 + ...
such that A(x) = 1 + x*G(x)^5 where
G(x) = 1 + 2*x + 16*x^2 + 180*x^3 + 2360*x^4 + 33760*x^5 + 510928*x^6 + 8043440*x^7 + ... + A363311(n)*x^n + ...
satisfies G(x) = 1 + x*(G(x)^3 + G(x)^5).
Also, A(x) = B(x*A(x)^2) where B(x) = A(x/B(x)^2) begins
B(x) = 1 + x + 8*x^2 + 67*x^3 + 590*x^4 + 5403*x^5 + 51034*x^6 + 494268*x^7 + ... + A363309(n)*x^n + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A363309, A363311, A001764.

Programs

PARI

{a(n) = if(n==0,1, sum(k=0, n-1, 5*binomial(n-1, k)*binomial(3*n+2*k+2, n-1)/(3*n+2*k+2) ) )}
for(n=0, 20, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined as follows; here, B(x) is the g.f. of A363309 and F(x) is the g.f. of A001764.

(1) A(x) = 1 + x*G(x)^5, where G(x) = 1 + x*(G(x)^3 + G(x)^5) is the g.f. of A363311.

(2) A(x) = B(x*A(x)^2) where B(x) = F(x*F(x)^5) and F(x) = 1 + x*F(x)^3.

(3) A(x) = sqrt( (1/x)*Series_Reversion( x/B(x)^2 ) ), where B(x) is the g.f. of A363309.

(4) a(n) = Sum_{k=0..n-1} 5*binomial(n-1, k)*binomial(3*n+2*k+2, n-1)/(3*n+2*k+2) for n > 1 with a(0) = 1.