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%I #14 Aug 04 2020 21:34:57
%S 0,0,0,0,0,1,1,0,0,0,0,1,1,2,1,0,0,1,1,2,3,2,1,1,1,2,3,5,4,4,2,2,3,5,
%T 7,8,7,5,4,5,7,12,12,13,11,9,8,12,16,19,21,20,17,16,17,24,28,33,33,33,
%U 28,29,34,41,48,54,53,50,48,52,58,72,79,85,83,81,80,88,100,116,127
%N Number of partitions of n into parts 8k+6 or 8k+7.
%C In general, if a > 0, b > 0, c > 0, d > 0, gcd(a,b) = 1, gcd(c,d) = 1 and g.f. = Product_{k>=0} 1/((1 - x^(a*k+b)) * (1 - x^(c*k+d))), then a(n) ~ Gamma(b/a) * Gamma(d/c) * a^((2*b/a - 2*d/c - 1)/4) * c^((2*d/c - 2*b/a - 1)/4) * (a+c)^((2*b/a + 2*d/c - 1)/4) * Pi^(b/a + d/c - 2) * exp(Pi*sqrt(2*(1/a + 1/c)*n/3)) / (2^((2*b/a + 2*d/c + 7)/4) * 3^((2*b/a + 2*d/c - 1)/4) * n^((1 + 2*b/a + 2*d/c)/4)). - _Vaclav Kotesovec_, Aug 27 2015
%H Robert Price, <a href="/A035468/b035468.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) ~ exp(Pi*sqrt(n/6)) * Gamma(3/4) * Gamma(7/8) / (2^(29/16) * 3^(9/16) * Pi^(3/8) * n^(17/16)). - _Vaclav Kotesovec_, Aug 27 2015
%t nmax = 100; Rest[CoefficientList[Series[Product[1/((1 - x^(8k+6))*(1 - x^(8k+7))), {k, 0, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Aug 27 2015 *)
%t nmax = 60; kmax = nmax/8;
%t s = Flatten[{Range[0, kmax]*8 + 6}~Join~{Range[0, kmax]*8 + 7}];
%t Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 1, nmax}] (* _Robert Price_, Aug 04 2020 *)
%K nonn
%O 1,14
%A _Olivier Gérard_
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