A166879 -id:A166879 - cp's OEIS frontend

A165937 G.f.: A(x) = exp( Sum_{n>=1} A002203(n^2)*x^n/n ).

1, 2, 19, 964, 334965, 742714950, 10042408885191, 814556580116590856, 393147641272746246076745, 1123539400297807898234860367690, 18948227277012085227250633551784337179, 1881331163508674280605070386666674939623268684

Offset: 0

Paul D. Hanna, Oct 18 2009

nonn

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

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

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

{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))}

Logarithmic derivative equals A165938.

Self-convolution of A166879.

A204327 a(n) = Pell(n^2).

Original entry on oeis.org

1, 12, 985, 470832, 1311738121, 21300003689580, 2015874949414289041, 1111984844349868137938112, 3575077977948634627394046618865, 66992092050551637663438906713182313772, 7316660981177400006023755031791634132229378601

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

			G.f.: A(x) = x + 12*x^2 + 985*x^3 + 470832*x^4 + 1311738121*x^5 +...

Seiichi Manyama, Table of n, a(n) for n = 1..51

Cf. A000129 (Pell), A000290 (n^2), A204274, A165938, A165937, A166879, A054783.

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{a(n)=Pell(n^2)}

a(n) = ( (1+sqrt(2))^(n^2) - (1-sqrt(2))^(n^2) ) / (2*sqrt(2)).

cp's OEIS Frontend

A165937 G.f.: A(x) = exp( Sum_{n>=1} A002203(n^2)*x^n/n ).

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