A208055 G.f.: exp( Sum_{n>=1} 2*Pell(n)^4 * x^n/n ), where Pell(n) = A000129(n).

1, 2, 18, 450, 11362, 311426, 8857426, 259072706, 7730804098, 234255654466, 7184570715602, 222512186923010, 6947171244623714, 218374183252085826, 6903938704875627410, 219355658720815861378, 6999679608428089841154, 224210965624588803552642

Offset: 0

Author

Paul D. Hanna, Feb 22 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 450*x^3 + 11362*x^4 + 311426*x^5 +...
such that, by definition,
log(A(x))/2 = x + 2^4*x^2/2 + 5^4*x^3/3 + 12^4*x^4/4 + 29^4*x^5/5 + 70^4*x^6/6 + 169^4*x^7/7 + 408^4*x^8/8 +...+ Pell(n)^4*x^n/n +...
		

Crossrefs

Cf. A000129, A208034, A208056, A204061, A204062, A207969.

Programs

PARI

{Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(m=1,n,2*Pell(m)^4*x^m/m) +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

Formula

The o.g.f. A(x) = 1 + 2*x + 18*x^2 + 450*x^3 + ... is an algebraic function: A(x)^32 = (1 + 6*x + x^2)^4/( (1 - 34*x + x^2)*(1 - 2*x + x^2)^3 ). Cf. A207969. - Peter Bala, Apr 03 2014

From Vaclav Kotesovec, Oct 31 2024: (Start)

G.f.: (1 + x*(6 + x))^(1/8) / ((1 - x)^(3/16)*(1 + (-17 + 12*sqrt(2))*x)^(1/32) * (1 - (17 + 12*sqrt(2))*x)^(1/32)).

a(n) ~ 5^(1/8) * (1 + sqrt(2))^(4*n) / (2^(13/64) * 3^(1/32) * Gamma(1/32) * n^(31/32)). (End)