A215171 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

A215171

G.f.: exp( Sum_{n>=1} A002203(n)^4 * x^n/n ), where A002203 is the companion Pell numbers.

1, 16, 776, 23856, 834596, 28135056, 957599096, 32515276336, 1104679254346, 37525681919856, 1274775209167896, 43304782313176656, 1471088177488196276, 49973690736096892016, 1697634414511896630376, 57669596280038205388752, 1959068639950002397935907

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

More generally, exp(Sum_{k>=1} A002203(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A002203(2*k)*x - x^2)^binomial(2*n,n-k).

Compare to g.f. exp(Sum_{k>=1} A002203(k) * x^k/k) = 1/(1-2*x-x^2).

LINKS

Table of n, a(n) for n=0..16.

FORMULA

G.f.: 1/((1-x)^6*(1+6*x+x^2)^4*(1-34*x+x^2)).

EXAMPLE

G.f.: A(x) = 1 + 16*x + 776*x^2 + 23856*x^3 + 834596*x^4 + 28135056*x^5 +...

where

log(A(x)) = 2^4*x + 6^4*x^2/2 + 14^4*x^3/3 + 34^4*x^4/4 + 82^4*x^5/5 + 198^4*x^6/6 + 478^4*x^7/7 + 1154^4*x^8/8 +...+ A002203(n)^4*x^n/n +...

PROG

(PARI) {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)), n)}

{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^4*x^k/k)+x*O(x^n)), n)}

(PARI) {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)), n)}

{a(n, m=2)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m, m) * prod(k=1, m, 1/(1 - (-1)^(m-k)*A002203(2*k)*x + x^2+x*O(x^n))^binomial(2*m, m-k)), n)}

CROSSREFS

Cf. A204062, A212442, A203804.

Sequence in context: A221061 A326216 A194610 * A224736 A374606 A232519

Adjacent sequences: A215168 A215169 A215170 * A215172 A215173 A215174

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 05 2012

STATUS

approved