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A280454 Expansion of Product_{k>=0} (1 + x^(5*k+1)).

%I A280454 #14 Jan 24 2017 09:47:11
%S A280454 1,1,0,0,0,0,1,1,0,0,0,1,1,0,0,0,1,2,1,0,0,1,2,1,0,0,1,3,2,0,0,1,3,3,
%T A280454 1,0,1,4,4,1,0,1,4,5,2,0,1,5,7,3,0,1,5,8,5,1,1,6,10,6,1,1,6,12,9,2,1,
%U A280454 7,14,11,3,1,7,16,15,5,1,8,19,18,7,2,8,21,23,10,2,9,24,27,13,3,9,27,34,18,4,10,30,39,23,6,10
%N A280454 Expansion of Product_{k>=0} (1 + x^(5*k+1)).
%C A280454 Number of partitions of n into distinct parts congruent to 1 mod 5.
%C A280454 Convolution of A281243 and A280454 is A203776. - _Vaclav Kotesovec_, Jan 18 2017
%H A280454 Vaclav Kotesovec, <a href="/A280454/b280454.txt">Table of n, a(n) for n = 0..10000</a>
%H A280454 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015.
%H A280454 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A280454 G.f.: Product_{k>=0} (1 + x^(5*k+1)).
%F A280454 a(n) ~ exp(Pi*sqrt(n)/sqrt(15))/(2*2^(1/5)*15^(1/4)*n^(3/4)) * (1 + (Pi/(240*sqrt(15)) - 3*sqrt(15)/(8*Pi)) / sqrt(n)). - _Ilya Gutkovskiy_, Jan 03 2017, extended by _Vaclav Kotesovec_, Jan 24 2017
%e A280454 a(27) = 3 because we have [26, 1], [21, 6] and [11, 16].
%t A280454 nmax = 102; CoefficientList[Series[Product[(1 + x^(5 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
%t A280454 nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 5] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Jan 18 2017 *)
%Y A280454 Cf. A000700, A016861, A109697, A169975, A261612, A280456, A280457.
%Y A280454 Cf. A262928, A147599, A281243, A281244, A281245.
%K A280454 nonn
%O A280454 0,18
%A A280454 _Ilya Gutkovskiy_, Jan 03 2017