A322241 G.f.: exp( Sum_{n>=1} A084605(n)^2 * x^n/n ), where A084605(n) is the central coefficient in (1 + x + 4*x^2)^n.

1, 1, 41, 249, 6305, 77569, 1665321, 27724889, 574252417, 10958980929, 228679916905, 4671350051321, 99292476904609, 2107949882690241, 45658568907254505, 993562984208479193, 21876513296218002433, 484448162130512673665, 10812975015547281792937, 242647271141110287979513, 5477046865641884201456033

Offset: 0

Author

Paul D. Hanna, Dec 08 2018

Keywords

• nonn

Comments

Compare to: exp( Sum_{n>=1} A084605(n) * x^n/n ) = (1-x - sqrt(1 - 2*x - 15*x^2))/(8*x^2), the g.f. of A091147.

Sequence A322240(n) = A084605(n)^2 has generating function 1 / AGM(1 + 15*x, sqrt((1 - 9*x)*(1 - 25*x)) ).

Examples

			G.f.: A(x) = 1 + x + 41*x^2 + 249*x^3 + 6305*x^4 + 77569*x^5 + 1665321*x^6 + 27724889*x^7 + 574252417*x^8 + 10958980929*x^9 + 228679916905*x^10 + ...
such that
log(A(x)) = x + 81*x^2/2 + 625*x^3/3 + 21025*x^4/4 + 314721*x^5/5 + 8071281*x^6/6 + 155975121*x^7/7 + 3685097025*x^8/8 + ... + A084605(n)^2 * x^n/n + ...
RELATED SERIES.
The g.f. of A084605 equals the series
1/sqrt(1 - 2*x - 15*x^2) = 1 + x + 9*x^2 + 25*x^3 + 145*x^4 + 561*x^5 + 2841*x^6 + 12489*x^7 + 60705*x^8 + 281185*x^9 + ... + A084605(n) * x^n/n + ...
		

Crossrefs

Cf. A322240, A084605.

Programs

PARI

{a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, polcoeff(1/sqrt(1 - 2*x - 15*x^2 +x*O(x^m)), m)^2 *x^m/m)+x*O(x^n)), n))}
for(n=0,30,print1(a(n),", "))