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A209445 a(n) = Pell(n)*A001227(n) for n >= 1, where A001227(n) is the number of odd divisors of n.

%I A209445 #12 Jan 05 2018 03:00:26
%S A209445 1,2,10,12,58,140,338,408,2955,4756,11482,27720,66922,161564,780100,
%T A209445 470832,2273378,8232630,13250218,31988856,154455860,186444716,
%U A209445 450117362,1086679440,3935214363,6333631924,30581480180,36915112104,89120964298,430314081400,519435045698
%N A209445 a(n) = Pell(n)*A001227(n) for n >= 1, where A001227(n) is the number of odd divisors of n.
%C A209445 Compare g.f. to the Lambert series of A001227: Sum_{n>=1} x^(2*n-1)/(1 - x^(2*n-1)).
%H A209445 G. C. Greubel, <a href="/A209445/b209445.txt">Table of n, a(n) for n = 1..1000</a>
%F A209445 G.f.: Sum_{n>=1} Pell(2*n-1)*x^(2*n-1)/(1 - A002203(2*n-1)*x^(2*n-1)-x^(4*n-2)), where A002203(n) = Pell(n-1) + Pell(n+1).
%e A209445 G.f.: A(x) = x + 2*x^2 + 10*x^3 + 12*x^4 + 58*x^5 + 140*x^6 + 338*x^7 + ...
%e A209445 where A(x) = 1*1*x + 2*1*x^2 + 5*2*x^3 + 12*1*x^4 + 29*2*x^5 + 70*2*x^6 + 169*2*x^7 + 408*1*x^8 + ... + Pell(n)*A001227(n)*x^n + ...
%e A209445 The g.f. is also given by the identity:
%e A209445 A(x) = 1*x/(1-2*x-x^2) + 5*x^3/(1-14*x^3-x^6) + 29*x^5/(1-82*x^5-x^10) + 169*x^7/(1-478*x^7-x^14) + 985*x^9/(1-2786*x^9-x^18) + 5741*x^11/(1-16238*x^11-x^22) + ...
%e A209445 which involves odd-indexed Pell and A002203 numbers.
%t A209445 A001227[n_]:= Sum[Mod[d, 2], {d, Divisors[n]}]; Table[Fibonacci[n, 2]*A001227[n], {n, 1, 1000}] (* _G. C. Greubel_, Jan 02 2018 *)
%o A209445 (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
%o A209445 {A002203(n)=Pell(n-1)+Pell(n+1)}
%o A209445 {a(n)=polcoeff(sum(m=1,n,Pell(2*m-1)*x^(2*m-1)/(1-A002203(2*m-1)*x^(2*m-1)-x^(4*m-2)+x*O(x^n))),n)}
%o A209445 for(n=1,40,print1(a(n),", "))
%Y A209445 Cf. A001227, A205965, A209444, A209446, A204270.
%K A209445 nonn
%O A209445 1,2
%A A209445 _Paul D. Hanna_, Mar 09 2012