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

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