A225524 - OEIS

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

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OFFSET

0,2

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).

LINKS

Table of n, a(n) for n=0..31.

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.

EXAMPLE

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) *...

PROG

(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), ", "))

CROSSREFS

Cf. A203850, A225525.