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A280457 Expansion of Product_{k>=0} (1 + x^(7*k+1)).

%I A280457 #14 Jan 24 2017 09:53:54
%S A280457 1,1,0,0,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,0,0,0,1,2,1,0,0,0,0,1,2,1,0,0,
%T A280457 0,0,1,3,2,0,0,0,0,1,3,3,1,0,0,0,1,4,4,1,0,0,0,1,4,5,2,0,0,0,1,5,7,3,
%U A280457 0,0,0,1,5,8,5,1,0,0,1,6,10,6,1,0,0,1,6,12,9,2,0,0,1,7,14,11,3,0,0,1,7,16,15,5
%N A280457 Expansion of Product_{k>=0} (1 + x^(7*k+1)).
%C A280457 Number of partitions of n into distinct parts congruent to 1 mod 7.
%H A280457 Vaclav Kotesovec, <a href="/A280457/b280457.txt">Table of n, a(n) for n = 0..10000</a>
%H A280457 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 A280457 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A280457 G.f.: Product_{k>=0} (1 + x^(7*k+1)).
%F A280457 a(n) ~ exp(Pi*sqrt(n)/sqrt(21))/(2*2^(1/7)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(336*sqrt(21)) - 3*sqrt(21)/(8*Pi)) / sqrt(n)). - _Ilya Gutkovskiy_, Jan 03 2017, extended by _Vaclav Kotesovec_, Jan 24 2017
%e A280457 a(37) = 3 because we have [36, 1], [29, 8] and [22, 15].
%t A280457 nmax = 105; CoefficientList[Series[Product[(1 + x^(7 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
%t A280457 nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 7] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Jan 18 2017 *)
%Y A280457 Cf. A000700, A016993, A169975, A261612, A280454.
%Y A280457 Cf. A262928, A147599, A281243, A281244.
%Y A280457 Cf. A109703, A281245, A281455, A281456, A281457, A281458, A281459.
%K A280457 nonn
%O A280457 0,24
%A A280457 _Ilya Gutkovskiy_, Jan 03 2017