A261770 - OEIS

%I #15 Dec 07 2017 22:19:56

%S 1,1,1,2,2,3,3,4,5,6,8,9,11,13,16,19,22,26,30,35,41,47,55,63,73,84,96,

%T 110,125,143,162,184,208,235,266,300,338,380,427,479,536,600,670,748,

%U 834,929,1034,1149,1277,1417,1571,1740,1925,2129,2351,2596,2863

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

%C a(n) is the number of partitions of n into distinct parts where no part is a multiple of 6. - _Joerg Arndt_, Aug 31 2015

%H Alois P. Heinz, <a href="/A261770/b261770.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, p. 12.

%F a(n) ~ exp(Pi*sqrt(5*n/2)/3) * 5^(1/4) / (2^(7/4) * sqrt(3) * n^(3/4)) * (1 - (9/(4*Pi*sqrt(10)) + 5*Pi*sqrt(5/2)/144) / sqrt(n)). - _Vaclav Kotesovec_, Aug 31 2015, extended Jan 21 2017

%F G.f.: Product_{k>=1} (1 - x^(12*k-6))/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 07 2017

%p b:= proc(n, i) option remember;  local r;

%p       `if`(2*n>i*(i+1)-(j-> 6*j*(j+1))(iquo(i, 6, 'r')), 0,

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

%p     end:

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

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, Aug 31 2015

%t nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(6*k)), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A261736.

%Y Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A097793 (m=7), A261771 (m=8), A112193 (m=9), A261772 (m=10).

%Y Column k=6 of A290307.

%K nonn

%O 0,4

%A _Vaclav Kotesovec_, Aug 31 2015