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 A261568 Expansion of Product_{k>=1} (1 + 4*x^k). 8

%I

%S 1,4,4,20,20,36,100,116,180,260,580,660,1044,1380,2020,3444,4340,6020,

%T 8260,11220,14740,23140,28196,38900,50420,67780,85956,114900,157140,

%U 197860,257060,331060,423540,540100,687620,864084,1145300,1406500,1789860,2231860

%N Expansion of Product_{k>=1} (1 + 4*x^k).

%H Vaclav Kotesovec, <a href="/A261568/b261568.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(5*Pi)*n^(3/4)), where c = Pi^2/6 + 2*log(2)^2 + polylog(2, -1/4) = 2.36993979699836583198553742535032304875... . - _Vaclav Kotesovec_, Jan 04 2016

%F G.f.: Sum_{i>=0} 4^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - _Ilya Gutkovskiy_, Apr 12 2018

%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,

%p `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, 4*b(n-i, i-1))))

%p end:

%p a:= n-> b(n\$2):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Aug 24 2015

%t nmax = 40; CoefficientList[Series[Product[1 + 4*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*4^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 25 2015 *)

%t (QPochhammer[-4, x]/5 + O[x]^58)[[3]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)

%Y Cf. A000009, A032302, A032308, A261569.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Aug 24 2015

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Last modified December 8 14:38 EST 2019. Contains 329865 sequences. (Running on oeis4.)