A280456 - OEIS

%I #12 Jan 18 2017 13:05:19

%S 1,1,0,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,1,2,1,0,0,0,1,2,1,0,0,0,1,3,2,

%T 0,0,0,1,3,3,1,0,0,1,4,4,1,0,0,1,4,5,2,0,0,1,5,7,3,0,0,1,5,8,5,1,0,1,

%U 6,10,6,1,0,1,6,12,9,2,0,1,7,14,11,3,0,1,7,16,15,5,0,1,8,19,18,7,1,1,8,21,23,10,1,1,9,24

%N Expansion of Product_{k>=0} (1 + x^(6*k+1)).

%C Number of partitions of n into distinct parts congruent to 1 mod 6.

%C Convolution of A281244 and A280456 is A098884. - _Vaclav Kotesovec_, Jan 18 2017

%H Vaclav Kotesovec, <a href="/A280456/b280456.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>

%F G.f.: Product_{k>=0} (1 + x^(6*k+1)).

%F a(n) ~ exp(Pi*sqrt(n)/(3*sqrt(2)))/(2*2^(5/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/(144*sqrt(2)) - 9/(4*sqrt(2)*Pi)) / sqrt(n)). - _Ilya Gutkovskiy_, Jan 03 2017, extended by _Vaclav Kotesovec_, Jan 18 2017

%e a(32) = 3 because we have [31, 1], [25, 7] and [19, 13].

%t nmax = 105; CoefficientList[Series[Product[(1 + x^(6 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

%t nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 6] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Jan 18 2017 *)

%Y Cf. A000700, A016921, A109701, A169975, A261612, A280454, A280457.

%Y Cf. A262928, A147599, A281243, A281244, A281245.

%K nonn

%O 0,21

%A _Ilya Gutkovskiy_, Jan 03 2017