OFFSET
0,2
COMMENTS
Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), and to the g.f. of Pell numbers: exp( Sum_{n>=1} A002203(n)*x^n/n ).
FORMULA
EXAMPLE
G.f.: A(x) = 1 + 2*x + 11*x^2 + 38*x^3 + 156*x^4 + 516*x^5 + 1991*x^6 +...
where
A(x) = 1/((1-2*x-x^2) * (1-6*x^2+x^4) * (1-14*x^3-x^6) * (1-34*x^4+x^8) * (1-82*x^5-x^10) * (1-198*x^6+x^12) *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) *...).
The companion Pell numbers (starting at offset 1) begin:
A002203 = [2,6,14,34,82,198,478,1154,2786,6726,16238,...].
PROG
(PARI) /* Subroutine used in PARI programs below: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)), n)}
(PARI) {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}
(PARI) {a(n)=polcoeff(prod(m=1, n, 1/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
(PARI) {a(n)=if(n<0, 0, if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)*A002203(k)*a(n-k))))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 02 2012
STATUS
approved