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

%I A261051 #11 May 30 2018 15:40:50
%S A261051 1,1,3,7,14,33,69,148,307,642,1314,2684,5432,10924,21841,43431,85913,
%T A261051 169170,331675,647601,1259737,2441706,4716874,9083215,17439308,
%U A261051 33387589,63749174,121409236,230658963,437198116,826838637,1560410267,2938808875,5524005110
%N A261051 Expansion of Product_{k>=1} (1+x^k)^(Lucas(k)).
%H A261051 Vaclav Kotesovec, <a href="/A261051/b261051.txt">Table of n, a(n) for n = 0..1000</a>
%H A261051 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1508.01796">Asymptotics of the Euler transform of Fibonacci numbers</a>, arXiv:1508.01796 [math.CO], Aug 07 2015
%F A261051 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.
%F A261051 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
%p A261051 L:= n-> (<<0|1>, <1|1>>^n. <<2, 1>>)[1, 1]:
%p A261051 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A261051        add(binomial(L(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A261051     end:
%p A261051 a:= n-> b(n$2):
%p A261051 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 08 2015
%t A261051 nmax=40; CoefficientList[Series[Product[(1+x^k)^LucasL[k],{k,1,nmax}],{x,0,nmax}],x]
%Y A261051 Cf. A000032, A261031, A261050, A026007, A027998, A248882, A102866, A256142.
%K A261051 nonn
%O A261051 0,3
%A A261051 _Vaclav Kotesovec_, Aug 08 2015