A205508 - OEIS

%I #5 Mar 30 2012 18:37:34

%S 1,4,8,0,48,232,0,0,1632,3940,19024,0,0,267688,0,0,1883328,9093512,

%T 10976840,0,127955424,0,0,0,0,15740857452,25334527696,0,0,

%U 356483857192,0,0,2508054264192,0,29236023007504,0,85200014758320,411382062287848,0,0,5788584895037376

%N  a(n) = Pell(n) * A004018(n) for n>=1 with a(0)=1, where A004018(n) is the number of ways of writing n as a sum of 2 squares.

%C  Compare to the g.f. of A004018 given by the Lambert series identity:

%C 1 + 4*Sum_{n>=0} (-1)^n*x^(2*n+1)/(1 - x^(2*n+1)) = (1 + 2*Sum_{n>=1} x^(n^2))^2.

%F  G.f.: 1 + 4*Sum_{n>=0} (-1)^n*Pell(2*n+1)*x^(2*n+1) / (1 - A002203(2*n+1)*x^(2*n+1) - x^(4*n+2)), where A002203 is the companion Pell numbers.

%e  G.f.: A(x) = 1 + 4*x + 8*x^2 + 48*x^4 + 232*x^5 + 1632*x^8 + 3940*x^9 + 19024*x^10 +...

%e Compare the g.f to the square of the Jacobi theta_3 series:

%e theta_3(x)^2 = 1 + 4*x + 4*x^2 + 4*x^4 + 8*x^5 + 4*x^8 + 4*x^9 + 8*x^10 +...+ A004018(n)*x^n +...

%e The g.f. equals the sum:

%e A(x) = 1 + 4*x/(1-2*x-x^2) - 4*5*x^3/(1-14*x^3-x^6) + 4*29*x^5/(1-82*x^5-x^10) - 4*169*x^7/(1-478*x^7-x^14) + 4*985*x^9/(1-2786*x^9-x^18) - 4*5741*x^11/(1-16238*x^11-x^22) + 4*33461*x^13/(1-94642*x^13-x^26) - 4*195025*x^15/(1-551614*x^15-x^30) +...

%e which involves odd-indexed Pell and companion Pell numbers.

%o  (PARI) {A004018(n)=polcoeff((1+2*sum(k=1, sqrtint(n+1), x^(k^2), x*O(x^n)))^2, n)}

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

%o {a(n)=if(n==0,1,Pell(n)*A004018(n))}

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

%o {A002203(n)=Pell(n-1)+Pell(n+1)}

%o {a(n)=polcoeff((1+4*sum(m=0,n+1,(-1)^m*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))))^(1/1),n)}

%Y  Cf. A205507, A204384, A204270, A004018, A000129 (Pell), A002203.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 28 2012