A261050 Expansion of Product_{k>=1} (1+x^k)^(Fibonacci(k)).

1, 1, 1, 3, 5, 10, 19, 36, 67, 127, 236, 438, 811, 1496, 2750, 5046, 9224, 16827, 30630, 55623, 100803, 182342, 329205, 593326, 1067591, 1917885, 3440207, 6162004, 11021921, 19688757, 35126020, 62590629, 111398910, 198044551, 351700332, 623918086, 1105715149

Offset: 0

Author

Vaclav Kotesovec, Aug 08 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015

Crossrefs

Cf. A000045, A166861, A261051, A026007, A027998, A248882, A102866, A256142, A260916.

Programs

Maple

f:= proc(n) option remember; (<<1|1>, <1|0>>^n)[1, 2] end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(binomial(f(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 08 2015

Mathematica

nmax=40; CoefficientList[Series[Product[(1+x^k)^Fibonacci[k],{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ phi^n / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(-1/10 + 2*5^(-1/4)*sqrt(n) + s), where s = Sum_{k>=2} (-1)^(k+1) * phi^k / ((phi^(2*k) - phi^k - 1)*k) = -0.3237251774053525012502809827680337358578568068831886835557918847... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 29 2018