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

Author

Paul D. Hanna, Jan 02 2012

Keywords

• nonn

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 ).

Examples

			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,...].
		

Crossrefs

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

Programs

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))))}

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.