A165937 - OEIS

%I #8 Aug 16 2012 23:19:33

%S 1,2,19,964,334965,742714950,10042408885191,814556580116590856,

%T 393147641272746246076745,1123539400297807898234860367690,

%U 18948227277012085227250633551784337179,1881331163508674280605070386666674939623268684

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

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

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

%C Given g.f. A(x), (1-x)^(1/4) * A(x)^(1/8) is an integer series.

%F Logarithmic derivative equals A165938.

%F Self-convolution of A166879.

%e G.f.: A(x) = 1 + 2*x + 19*x^2 + 964*x^3 + 334965*x^4 + 742714950*x^5 +...

%e log(A(x)) = 2*x + 34*x^2/2 + 2786*x^3/3 + 1331714*x^4/4 + 3710155682*x^5/5 + 60245508192802*x^6/6 + 5701755387019728962*x^7/7 +...+ A002203(n^2)*x^n/n +...

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

%Y Cf. A165938, A166879, A002203, A000129; variants: A158843, A166168, A155200.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 18 2009