A207972 Expansion of g.f.: exp( Sum_{n>=1} 5*Fibonacci(n^2) * x^n/n ).

1, 5, 20, 115, 1665, 82650, 12847310, 5620114060, 6659421195205, 21082748688390045, 177217804775828062850, 3941798437750184226876305, 231505293200405380457355524620, 35848160499603817968830380832049915, 14619744406297572472084577939841875791890

Offset

0,2

Comments

Moss and Ward prove that this is an integral sequence. - Peter Bala, Nov 28 2022

Let A(x) be the g.f. for this sequence. Note that the expansion of A(x)^(1/5) = exp( Sum_{n>=1} Fibonacci(n^2) * x^n/n ) does not have integer coefficients.

Links

Table of n, a(n) for n=0..14.

Patrick Moss and Tom Ward, Fibonacci along even powers is (almost) realizable, arXiv:2011.13068 [math.NT], 2020; Fibonacci Quart. 60 (2022), no. 1, 40-47.

Example

G.f.: A(x) = 1 + 5*x + 20*x^2 + 115*x^3 + 1665*x^4 + 82650*x^5 + ...
such that
log(A(x))/5 = x + 3*x^2/2 + 34*x^3/3 + 987*x^4/4 + 75025*x^5/5 + 14930352*x^6/6 + 7778742049*x^7/7 + ... + Fibonacci(n^2)*x^n/n + ...

Prog

(PARI) {a(n)=polcoeff(exp(sum(k=1, n, 5*fibonacci(k^2)*x^k/k)+x*O(x^n)), n)}
for(n=0, 16, print1(a(n), ", "))

Crossrefs

Cf. A054888, A207969, A207970, A207971, A054783, A207834, A211892.

Sequence in context: A300490 A020039 A319489 * A117736 A258665 A028944

Adjacent sequences: A207969 A207970 A207971 * A207973 A207974 A207975

Keyword

nonn,easy

Author

Paul D. Hanna, Feb 22 2012

Status

approved