A166879 G.f.: A(x) = exp( Sum_{n>=1} A002203(n^2)/2*x^n/n ).

1, 1, 9, 473, 166969, 371186249, 5020831641761, 407273265807001089, 196573413317730320842177, 561769503571822735164882969633, 9474113076734769687535254457293566857, 940665572280219007549184269220597591870817337

Offset: 0

Author

Paul D. Hanna, Oct 22 2009

Keywords

• nonn

Comments

A002203 equals the logarithmic derivative of the Pell numbers (A000129).

Note that A002203(n^2) = (1+sqrt(2))^(n^2) + (1-sqrt(2))^(n^2).

Examples

			G.f.: A(x) = 1 + x + 9*x^2 + 473*x^3 + 166969*x^4 + 371186249*x^5 +...
log(A(x)) = x + 17*x^2/2 + 1393*x^3/3 + 665857*x^4/4 + 1855077841*x^5/5 + 30122754096401*x^6/6 + 2850877693509864481*x^7/7 +...+ A002203(n^2)/2*x^n/n +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..40

Crossrefs

Cf. A165937, A165938, A002203, A000129.

Programs

PARI

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,polcoeff((1-x)/(1-2*x-x^2+x*O(x^(m^2))),m^2)*x^m/m)+x*O(x^n)),n))}

PARI

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,polcoeff((1-x)/(1-2*x-x^2+x*O(x^(k^2))),k^2)*a(n-k)))}

Formula

a(n) == 1 (mod 8).

a(n) = (1/n)*Sum_{k=1..n} A002203(k^2)/2*a(n-k) for n>0 with a(0)=1.

Self-convolution yields A165937.