A203534 G.f.: exp( Sum_{n>=1} sigma(n)*A002203(n)*x^n/n ) where A002203 is the companion Pell numbers.

1, 2, 11, 38, 156, 516, 1991, 6434, 23065, 75132, 255335, 816480, 2724245, 8570794, 27763807, 87057596, 276339126, 855374534, 2681503010, 8218321006, 25421912010, 77383062314, 236519199902, 714226056554, 2165295121179, 6490447624984, 19503550719297, 58127246438024

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 ).

Links

Table of n, a(n) for n=0..27.

Formula

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

a(n) = (1/n)*Sum_{k=1..n} sigma(k)*A002203(k)*a(n-k) for n>0, with a(0) = 1.

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

Cf. A156234, A000129 (Pell), A002203 (companion Pell), A000203 (sigma).

Sequence in context: A294152 A196701 A196850 * A166989 A143550 A259213

Adjacent sequences: A203531 A203532 A203533 * A203535 A203536 A203537

Keyword

nonn

Author

Paul D. Hanna, Jan 02 2012

Status

approved