A207969 G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^4 * x^n/n ).

1, 5, 15, 60, 295, 1625, 9430, 56465, 345010, 2139595, 13419500, 84926105, 541398665, 3472389210, 22385362895, 144945232375, 942089445030, 6143582084115, 40181143112035, 263482860974570, 1731780213622125, 11406235045261205, 75268685723935940

Offset

0,2

Comments

Conjecture: exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k) * x^n/n ) is an integer series for integers k>=0.

Note that exp( Sum_{n>=1} 5*Fibonacci(n)^(2*k+1) * x^n/n ) is not an integer series for integers k.

Note that exp( Sum_{n>=1} Fibonacci(n)^(2*k) * x^n/n ) is not an integer series for integers k.

Links

Table of n, a(n) for n=0..22.

Formula

The o.g.f. A(x) = 1 + 5*x + 15*x^2 + 60*x^3 + ... is an algebraic function: A(x)^5 = (1 + 3*x + x^2)^4/( (1 - 7*x + x^2)*(1 - 2*x + x^2)^3 ). Cf. A203804. - Peter Bala, Apr 03 2014

a(n) ~ 2^(4/5) * 5^(1/10) * phi^(4*n) / (Gamma(1/5) * 3^(1/5) * n^(4/5)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 18 2020

Example

G.f.: A(x) = 1 + 5*x + 15*x^2 + 60*x^3 + 295*x^4 + 1625*x^5 + 9430*x^6 +...
such that
log(A(x))/5 = x + x^2/2 + 2^4*x^3/3 + 3^4*x^4/4 + 5^4*x^5/5 + 8^4*x^6/6 + 13^4*x^7/7 +...+ Fibonacci(n)^4*x^n/n +...

Prog

(PARI) {a(n)=polcoeff(exp(sum(k=1, n, 5*fibonacci(k)^4*x^k/k)+x*O(x^n)), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A054888, A207970, A207971, A207972, A207834, A207835.

A077916, A203804.

Sequence in context: A149599 A149600 A369555 * A149601 A149602 A149603

Adjacent sequences: A207966 A207967 A207968 * A207970 A207971 A207972

Keyword

nonn

Author

Paul D. Hanna, Feb 22 2012

Status

approved