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A203855 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^4, where Lucas(n) = A000204(n). 8
1, 40, 85, 580, 2928, 17440, 101040, 609660, 3706880, 22887192, 142567200, 895855380, 5667708960, 36072949560, 230763023408, 1482822818820, 9565561745040, 61920953016320, 402074969960400, 2618069854211784, 17090016552803440, 111812320834030800 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..400

FORMULA

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^5 * x^n/n), which is the g.f. of A203805.

a(n) ~ phi^(4*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

EXAMPLE

G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^40 * (1-4*x^3-x^6)^85 *

(1-7*x^4+x^8)^580 * (1-11*x^5-x^10)^2928 * (1-18*x^6+x^12)^17440 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) *...)

where F(x) = exp( Sum_{n>=1} Lucas(n)^5 * x^n/n ) = g.f. of A203805:

F(x) = 1 + x + 122*x^2 + 463*x^3 + 11985*x^4 + 85456*x^5 +...

where

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

MATHEMATICA

a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^4 &]; Array[a, 30] (* Jean-Fran├žois Alcover, Dec 04 2015 *)

PROG

(PARI) {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^4)/n)}

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

{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^5*x^m/m)+x*O(x^n))); if(n==1, 1, polcoeff(F*prod(k=1, n-1, (1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)), n)/Lucas(n))}

CROSSREFS

Cf. A203805, A203853, A203854, A203856, A203857, A203858, A203859, A203800.

Sequence in context: A141528 A160282 A243803 * A188335 A185718 A043414

Adjacent sequences:  A203852 A203853 A203854 * A203856 A203857 A203858

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 07 2012

STATUS

approved

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Last modified October 19 09:31 EDT 2021. Contains 348074 sequences. (Running on oeis4.)