A261051 Expansion of Product_{k>=1} (1+x^k)^(Lucas(k)).

1, 1, 3, 7, 14, 33, 69, 148, 307, 642, 1314, 2684, 5432, 10924, 21841, 43431, 85913, 169170, 331675, 647601, 1259737, 2441706, 4716874, 9083215, 17439308, 33387589, 63749174, 121409236, 230658963, 437198116, 826838637, 1560410267, 2938808875, 5524005110

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. A000032, A261031, A261050, A026007, A027998, A248882, A102866, A256142.

Programs

Maple

L:= n-> (<<0|1>, <1|1>>^n. <<2, 1>>)[1, 1]:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(binomial(L(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)^LucasL[k],{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ phi^n / (2*sqrt(Pi)*n^(3/4)) * exp(-1 + 1/(2*sqrt(5)) + 2*sqrt(n) + s), where s = Sum_{k>=2} (-1)^(k+1) * (2 + phi^k)/((phi^(2*k) - phi^k - 1)*k) = -0.590290697526802161885355317939144642488927381134222996704542... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 2*x^k)/(k*(1 - x^k - x^(2*k)))). - Ilya Gutkovskiy, May 30 2018