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A261772 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(10*k)).

%I A261772 #15 Dec 07 2017 22:20:12
%S A261772 1,1,1,2,2,3,4,5,6,8,9,11,14,16,20,24,28,33,40,46,54,64,74,86,100,115,
%T A261772 133,154,176,202,231,263,300,342,388,440,499,563,636,718,808,909,1022,
%U A261772 1146,1284,1439,1608,1797,2006,2236,2490,2772,3081,3422,3800,4212
%N A261772 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(10*k)).
%C A261772 a(n) is the number of partitions of n into distinct parts where no part is a multiple of 10.
%C A261772 In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)/(1 + x^(m*k)), then a(n) ~ exp(Pi*sqrt((m-1)*n/(3*m))) * (m-1)^(1/4) / (2^(3/2) * 3^(1/4) * m^(1/4) * n^(3/4)) * (1 - (3*sqrt(3*m)/(8*Pi*sqrt(m-1)) + (m-1)^(3/2)*Pi/(48*sqrt(3*m))) / sqrt(n)). - _Vaclav Kotesovec_, Aug 31 2015, extended Jan 21 2017
%H A261772 Alois P. Heinz, <a href="/A261772/b261772.txt">Table of n, a(n) for n = 0..10000</a>
%F A261772 a(n) ~ exp(Pi*sqrt(3*n/10)) * 3^(1/4) / (2^(7/4) * 5^(1/4) * n^(3/4)) * (1 - (sqrt(15)/(4*Pi*sqrt(2)) + 3*Pi*sqrt(3)/(16*sqrt(10))) / sqrt(n)). - _Vaclav Kotesovec_, Aug 31 2015, extended Jan 21 2017
%F A261772 G.f.: Product_{k>=1} (1 - x^(20*k-10))/(1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Dec 07 2017
%p A261772 b:= proc(n, i) option remember;  local r;
%p A261772       `if`(2*n>i*(i+1)-(j-> 10*j*(j+1))(iquo(i, 10, 'r')), 0,
%p A261772       `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
%p A261772     end:
%p A261772 a:= n-> b(n$2):
%p A261772 seq(a(n), n=0..80);  # _Alois P. Heinz_, Aug 31 2015
%t A261772 nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(10*k)), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A261772 Cf. A145707.
%Y A261772 Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A112193 (m=9).
%Y A261772 Column k=10 of A290307.
%K A261772 nonn
%O A261772 0,4
%A A261772 _Vaclav Kotesovec_, Aug 31 2015