A203850 G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-x^2)^n) / (1 + Lucas(n)*x^n + (-x^2)^n) where Lucas(n) = A000204(n).

1, -2, -4, 0, 14, 16, 0, 0, 4, -152, -188, 0, 0, -44, 0, 0, 4414, 5456, -4, 0, 1288, 0, 0, 0, 0, -335406, -414728, 0, 0, -97904, 0, 0, 4, 0, -8828, 0, 66770564, 82532956, 0, 0, 19483388, -304, 0, 0, 0, 1756816, 0, 0, 0, -34787592002, -42999828492, 0, 60508, -10150882544, 0, 0, 0, 0, -915304508, 0, 0, 796

Offset: 0

Author

Paul D. Hanna, Jan 07 2012

Keywords

• sign

Comments

Compare to: Product_{n>=1} (1-q^n)/(1+q^n) = 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*n)-sigma(n)) * q^n/n ).

Examples

			G.f.: A(x) = 1 - 2*x - 4*x^2 + 14*x^4 + 16*x^5 + 4*x^8 - 152*x^9 - 188*x^10 +...
-log(A(x)) = 2*x + 4*3*x^2/2 + 8*4*x^3/3 + 8*7*x^4/4 + 12*11*x^5/5 + 16*18*x^6/6 +...+ (sigma(2*n)-sigma(n))*Lucas(n)*x^n/n +...
Compare to the logarithm of Jacobi theta4 H(x) = 1 + 2*Sum_{n>=1} (-1)^n*x^(n^2):
-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 +...
The g.f. equals the product:
A(x) = (1-x-x^2)/(1+x-x^2) * (1-3*x^2+x^4)/(1+3*x^2+x^4) * (1-4*x^3-x^6)/(1+4*x^3-x^6) * (1-7*x^4+x^8)/(1+7*x^4+x^8) * (1-11*x^5-x^10)/(1+11*x^5-x^10) *...* (1 - Lucas(n)*x^n + (-x^2)^n)/(1 + Lucas(n)*x^n + (-x^2)^n) *...
Positions of zeros form A022544:
[3,6,7,11,12,14,15,19,21,22,23,24,27,28,30,31,33,35,38,39,42,43,44,...]
which are numbers that are not the sum of 2 squares.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A203860, A203861, A022544, A000204 (Lucas), A203318, A225524.

Programs

PARI

/* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

PARI

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

PARI

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

PARI

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

Formula

a(n) = 0 iff n is not the sum of 2 squares (A022544).

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

G.f.: exp( Sum_{n>=1} -(sigma(2*n)-sigma(n)) * Lucas(n) * x^n/n ) where Lucas(n) = A000204(n).