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A215171 G.f.: exp( Sum_{n>=1} A002203(n)^4 * x^n/n ), where A002203 is the companion Pell numbers.

%I A215171 #6 May 30 2025 11:01:18
%S A215171 1,16,776,23856,834596,28135056,957599096,32515276336,1104679254346,
%T A215171 37525681919856,1274775209167896,43304782313176656,
%U A215171 1471088177488196276,49973690736096892016,1697634414511896630376,57669596280038205388752,1959068639950002397935907
%N A215171 G.f.: exp( Sum_{n>=1} A002203(n)^4 * x^n/n ), where A002203 is the companion Pell numbers.
%C A215171 More generally, exp(Sum_{k>=1} A002203(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A002203(2*k)*x - x^2)^binomial(2*n,n-k).
%C A215171 Compare to g.f. exp(Sum_{k>=1} A002203(k) * x^k/k) = 1/(1-2*x-x^2).
%F A215171 G.f.: 1/((1-x)^6*(1+6*x+x^2)^4*(1-34*x+x^2)).
%e A215171 G.f.: A(x) = 1 + 16*x + 776*x^2 + 23856*x^3 + 834596*x^4 + 28135056*x^5 +...
%e A215171 where
%e A215171 log(A(x)) = 2^4*x + 6^4*x^2/2 + 14^4*x^3/3 + 34^4*x^4/4 + 82^4*x^5/5 + 198^4*x^6/6 + 478^4*x^7/7 + 1154^4*x^8/8 +...+ A002203(n)^4*x^n/n +...
%o A215171 (PARI) {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
%o A215171 {a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^4*x^k/k)+x*O(x^n)), n)}
%o A215171 (PARI) {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
%o A215171 {a(n, m=2)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m, m) * prod(k=1, m, 1/(1 - (-1)^(m-k)*A002203(2*k)*x + x^2+x*O(x^n))^binomial(2*m, m-k)), n)}
%Y A215171 Cf. A204062, A212442, A203804.
%K A215171 nonn
%O A215171 0,2
%A A215171 _Paul D. Hanna_, Aug 05 2012