A212442 G.f.: exp( Sum_{n>=1} A002203(n)^3 * x^n/n ), where A002203 is the companion Pell numbers.

1, 8, 140, 1864, 26602, 373080, 5253564, 73911192, 1040045475, 14634444720, 205922568360, 2897549559600, 40771618763540, 573700205699920, 8072574516567400, 113589743388536528, 1598328982089075749, 22490195492277648120, 316461065874934143252

Offset

0,2

Comments

More generally, exp(Sum_{k>=1} A002203(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A002203(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..865

Index entries for linear recurrences with constant coefficients, signature (8,76,136,-38,-136,76,-8,-1).

Formula

G.f.: 1 / ( (1+2*x-x^2)^3 * (1-14*x-x^2) ).

G.f.: 1 / Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^A212443(n) where A212443(n) = (1/n)*Sum_{d|n} moebius(n/d)*A002203(d)^2.

a(0)=1, a(1)=8, a(2)=140, a(3)=1864, a(4)=26602, a(5)=373080, a(6)=5253564, a(7)=73911192, a(n) = 8*a(n-1) + 76*a(n-2) + 136*a(n-3) - 38*a(n-4) - 136*a(n-5) + 76*a(n-6) - 8*a(n-7) - a(n-8). - Harvey P. Dale, Feb 15 2015

Example

G.f.: A(x) = 1 + 8*x + 140*x^2 + 1864*x^3 + 26602*x^4 + 373080*x^5 + ...
where
log(A(x)) = 2^3*x + 6^3*x^2/2 + 14^3*x^3/3 + 34^3*x^4/4 + 82^3*x^5/5 + 198^3*x^6/6 + 478^3*x^7/7 + 1154^3*x^8/8 + ... + A002203(n)^3*x^n/n + ...
Also, the g.f. equals the infinite product:
A(x) = 1/( (1-2*x-x^2)^4 * (1-6*x^2+x^4)^16 * (1-14*x^3-x^6)^64 * (1-34*x^4+x^8)^280 * (1-82*x^5-x^10)^1344 * (1-198*x^6+x^12)^6496 * ... * (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^A212443(n) * ...).
The exponents in these products begin:
A212443 = [4, 16, 64, 280, 1344, 6496, 32640, 166320, 862400, ...].
The companion Pell numbers begin (at offset 1):
A002203 = [2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...].

Mathematica

CoefficientList[Series[1/((1+2x-x^2)^3(1-14x-x^2)), {x, 0, 30}], x] (* or *) LinearRecurrence[{8, 76, 136, -38, -136, 76, -8, -1}, {1, 8, 140, 1864, 26602, 373080, 5253564, 73911192}, 30] (* Harvey P. Dale, Feb 15 2015 *)

Prog

(PARI) /* Subroutine for the PARI programs that follow: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)), n)}
(PARI) /* G.F. by Definition: */
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^3*x^k/k)+x*O(x^n)), n)}
(PARI) /* G.F. as a Finite Product: */
{a(n, m=1)=polcoeff(prod(k=0, m, 1/(1 - (-1)^(m-k)*A002203(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1, m-k)), n)}
(PARI) /* G.F. as an Infinite Product: */
{A212443(n)=(1/n)*sumdiv(n, d, moebius(n/d)*A002203(d)^2)}
{a(n)=polcoeff(1/prod(m=1, n, (1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))^A212443(m)), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A212443, A203803, A002203, A204062.

Sequence in context: A345650 A092703 A234353 * A185248 A228867 A224735

Adjacent sequences: A212439 A212440 A212441 * A212443 A212444 A212445

Keyword

nonn

Author

Paul D. Hanna, May 17 2012

Status

approved