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A207970
G.f.: exp( Sum_{n>=1} 5*Fibonacci(n)^6 * x^n/n ).
5
1, 5, 15, 140, 1505, 21875, 319620, 4936985, 77358485, 1236083870, 19982821875, 326511608255, 5379199407890, 89249496596015, 1489580814490755, 24988546214618750, 421055477328447620, 7122346563647277860, 120891417096833214485, 2058225554792946621495
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.
FORMULA
The o.g.f. A(x) = 1 + 5*x + 15*x^2 + 140*x^3 + ... is an algebraic function: A(x)^25 = ( (1 + 2*x + x^2)^10*(1 + 7*x + x^2)^6 )/( (1 - 3*x + x^2)^15*(1 - 18*x + x^2) ). Cf. A203806. - Peter Bala, Apr 03 2014
a(n) ~ 2^(17/25) * 5^(13/50) * phi^(6*n) / (Gamma(1/25) * 3^(3/5) * n^(24/25)), 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 + 140*x^3 + 1505*x^4 + 21875*x^5 + 319620*x^6 + ...
such that
log(A(x))/5 = x + x^2/2 + 2^6*x^3/3 + 3^6*x^4/4 + 5^6*x^5/5 + 8^6*x^6/6 + 13^6*x^7/7 + ... + Fibonacci(n)^6*x^n/n + ...
PROG
(PARI) {a(n)=polcoeff(exp(sum(k=1, n, 5*fibonacci(k)^6*x^k/k)+x*O(x^n)), n)}
for(n=0, 31, print1(a(n), ", "))
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 22 2012
STATUS
approved