A204384 - OEIS

%I #10 Mar 30 2012 18:37:34

%S 1,-4,-4,0,68,56,0,0,4,-5572,-4616,0,0,-328,0,0,2663428,2206456,-4,0,

%T 156808,0,0,0,0,-7420309452,-6147187208,0,0,-436867144,0,0,4,0,

%U -5326856,0,120491016385604,99818026262072,0,0,7093848711176,-11144,0,0,0,86497488056,0,0,0

%N G.f.: Product_{n>=1} (1 - A002203(n)*x^n + (-x^2)^n) / (1 + A002203(n)*x^n + (-x^2)^n) where A002203(n) is the companion Pell numbers.

%C a(A022544(n)) = 0 where A022544 lists numbers that are not the sum of 2 squares.

%C Compare to: Product_{n>=1} (1-q^k)/(1+q^k) = 1 + 2*Sum_{n>=1} (-1)^n*q^(n^2), the Jacobi theta_4 function, which has the g.f: exp( Sum_{n>=1} -(sigma(2*k)-sigma(k)) * x^n/n ).

%H Paul D. Hanna, <a href="/A204384/b204384.txt">Table of n, a(n) for n = 0..500</a>

%F G.f.: Product_{n>=1} (1 - A002203(2*n-1)*x^(2*n-1) - x^(4*n-2))^2 * (1 - A002203(2*n)*x^(2*n) + x^(4*n)).

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

%e G.f.: A(x) = 1 - 4*x - 4*x^2 + 68*x^4 + 56*x^5 + 4*x^8 - 5572*x^9 - 4616*x^10 +...

%e -log(A(x)) = 2*2*x + 4*6*x^2/2 + 8*14*x^3/3 + 8*34*x^4/4 + 12*82*x^5/5 + 16*198*x^6/6 +...+ (sigma(2*n)-sigma(n))*A002203(n)*x^n/n +...

%e Compare to the logarithm of Jacobi theta4 H(x) = 1 + 2*Sum_{n>=1} (-1)^n*q^(n^2):

%e -log(H(x)) = 2*x + 4*x^2/2 + 8*x^3/3 + 8*x^4/4 + 12*x^5/5 + 16*x^6/6 + 16*x^7/7 +...+ (sigma(2*n)-sigma(n))*x^n/n +...

%e The g.f. equals the products:

%e A(x) = (1-2*x-x^2)/(1+2*x-x^2) * (1-6*x^2+x^4)/(1+6*x^2+x^4) * (1-14*x^3-x^6)/(1+14*x^3-x^6) * (1-34*x^4+x^8)/(1+34*x^4+x^8) * (1-82*x^5-x^10)/(1+82*x^5-x^10) *...* (1 - A002203(n)*x^n + (-x^2)^n)/(1 + A002203(n)*x^n + (-x^2)^n) *...

%e A(x) = (1-2*x-x^2)^2 * (1-6*x^2+x^4) * (1-14*x^3-x^6)^2 * (1-34*x^4+x^8) * (1-82*x^5-x^10)^2 *(1-198*x^6+x^12) * (1-478*x^7-x^14)^2 * (1-1154*x^8+x^16) *...

%e Positions of zeros form A022544:

%e [3,6,7,11,12,14,15,19,21,22,23,24,27,28,30,31,33,35,38,39,42,43,44,...]

%e which are numbers that are not the sum of 2 squares.

%o (PARI) /* Subroutine used in PARI programs below: */

%o {A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

%o (PARI) {a(n)=polcoeff(prod(m=1, n, 1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))/prod(m=1, n, 1 + A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

%o (PARI) {a(n)=polcoeff(prod(m=1, n\2+1, (1 - A002203(2*m-1)*x^(2*m-1) - x^(4*m-2))^2*(1 - A002203(2*m)*x^(2*m) + x^(4*m) +x*O(x^n))), n)}

%o (PARI) {a(n)=polcoeff(exp(sum(k=1, n,-(sigma(2*k)-sigma(k))*A002203(k)*x^k/k)+x*O(x^n)), n)}

%Y Cf. A203850, A204382, A204383, A022544.

%K sign

%O 0,2

%A _Paul D. Hanna_, Jan 14 2012