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%I #16 May 10 2018 09:39:27
%S 1,1,1,3,2,4,6,7,8,13,16,18,26,29,38,49,58,68,90,101,125,156,181,214,
%T 263,304,358,435,505,589,701,812,939,1115,1275,1485,1736,1991,2286,
%U 2667,3038,3489,4028,4588,5240,6036,6833,7787,8904,10078,11429,13020,14698
%N Expansion of Product_{k>=1} (1 + x^k + x^(3*k)).
%H Vaclav Kotesovec, <a href="/A264905/b264905.txt">Table of n, a(n) for n = 0..5000</a>
%F a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) + exp(-3*x)) dx = 0.9953865985263189816963357718655148864441174218433250148867... . - _Vaclav Kotesovec_, Jan 05 2016
%t nmax = 100; CoefficientList[Series[Product[1+x^k+x^(3*k), {k,1,nmax}], {x,0,nmax}], x]
%t nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[4]] = 1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] + If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* _Vaclav Kotesovec_, May 10 2018 *)
%Y Cf. A000009, A000726, A001935, A100405, A266647, A266648, A266649, A266650, A266686, A275820.
%K nonn
%O 0,4
%A _Vaclav Kotesovec_, Nov 28 2015
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