A225524 G.f.: exp( Sum_{n>=1} (sigma(2*n) - sigma(n))*Lucas(n)*x^n/n ), where Lucas(n) = A000204(n) and sigma(n) = A000203(n) is the sum of divisors of n.

1, 2, 8, 24, 66, 184, 488, 1248, 3136, 7776, 18780, 44880, 105896, 246124, 567008, 1293840, 2920626, 6545352, 14555388, 32115120, 70421792, 153451488, 332314512, 715843200, 1534016392, 3270661294, 6941489040, 14667591672, 30859685088, 64670865304, 135011595856, 280813639680

Offset: 0

Author

Paul D. Hanna, May 09 2013

Keywords

• nonn

Comments

Compare g.f. to theta_4(x) = exp( Sum_{n>=1} -(sigma(2*n)-sigma(n))*x^n/n ), where Jacobi theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n*x^(n^2).

Examples

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 24*x^3 + 66*x^4 + 184*x^5 + 488*x^6 + 1248*x^7 +...
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) *...
		

Crossrefs

Cf. A203850, A225525.

Programs

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{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)}
for(n=0,30,print1(a(n),", "))

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1/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)}
for(n=0,30,print1(a(n),", "))

PARI

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(exp(sum(k=1, n, (sigma(2*k)-sigma(k))*Lucas(k)*x^k/k)+x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

Formula

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

G.f.: 1/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)).

Logarithmic derivative equals A225525.