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A203806 G.f.: exp( Sum_{n>=1} A000204(n)^6 * x^n/n ) where A000204 is the Lucas numbers. 10
1, 1, 365, 1730, 97390, 948562, 26292937, 370813165, 7716851405, 127699557640, 2397734250216, 42004273130216, 763345960355450, 13608990417046650, 245008471017094450, 4389301146029065420, 78826300825689660420, 1413927351334191841100, 25376664633745265522450 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..795

FORMULA

G.f.: 1/( (1+x)^20 * (1-3*x+x^2)^15 * (1+7*x+x^2)^6 * (1-18*x+x^2) ).

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203856(n) where A203856(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^5.

EXAMPLE

G.f.: A(x) = 1 + x + 365*x^2 + 1730*x^3 + 97390*x^4 + 948562*x^5 + ...

where

log(A(x)) = x + 3^6*x^2/2 + 4^6*x^3/3 + 7^6*x^4/4 + 11^6*x^5/5 + 18^6*x^6/6 + 29^6*x^7/7 + 47^6*x^8/8 + ... + Lucas(n)^6*x^n/n + ...

MATHEMATICA

CoefficientList[Series[1/((1 + x)^20*(1 - 3*x + x^2)^15*(1 + 7*x + x^2)^6*(1 - 18*x + x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 25 2017 *)

PROG

(PARI) /* Subroutine used in PARI programs below: */

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}

(PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^6*x^k/k)+x*O(x^n)), n)}

(PARI) {a(n, m=3)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m, m) * prod(k=1, m, 1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m, m-k)), n)}

CROSSREFS

Cf. A002571, A203803, A203804, A203805, A203807, A203808, A203809.

Cf. A203856, A203800.

Sequence in context: A275271 A062681 A259948 * A251314 A275182 A275130

Adjacent sequences:  A203803 A203804 A203805 * A203807 A203808 A203809

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 06 2012

STATUS

approved

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Last modified December 7 22:50 EST 2021. Contains 349590 sequences. (Running on oeis4.)