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A203534 G.f.: exp( Sum_{n>=1} sigma(n)*A002203(n)*x^n/n ) where A002203 is the companion Pell numbers. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 6 09:44 EDT 2020. Contains 334860 sequences. (Running on oeis4.)