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A035460 Number of partitions of n into parts 8k+3 or 8k+5.

%I A035460 #23 Aug 04 2020 12:32:37
%S A035460 1,0,0,1,0,1,1,0,1,1,1,2,1,2,2,2,3,2,3,4,3,5,5,4,7,6,7,9,7,10,11,10,
%T A035460 14,13,14,18,16,20,22,21,27,26,29,34,32,39,41,41,51,49,54,63,60,71,76,
%U A035460 76,90,91,98,111,110,125,133,137,157,159,172,191,192,216,229,235,266,272
%N A035460 Number of partitions of n into parts 8k+3 or 8k+5.
%H A035460 G. C. Greubel, <a href="/A035460/b035460.txt">Table of n, a(n) for n = 0..1000</a>
%F A035460 Expansion of f(-x^8) / f(-x^3, -x^5) in powers of x where f() is a Ramanujan theta function. - _Michael Somos_, Jun 03 2014
%F A035460 Euler transform of period 8 sequence [ 0, 0, 1, 0, 1, 0, 0, 0, ...]. - _Michael Somos_, Jun 03 2014
%F A035460 a(n) ~ (3-2*sqrt(2))^(1/4) * exp(Pi*sqrt(n/6)) / (4 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 26 2015
%e A035460 G.f. = 1 + x^3 + x^5 + x^6 + x^8 + x^9 + x^10 + 2*x^11 + x^12 + 2*x^13 + ...
%t A035460 a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ x^3, x^8] / QPochhammer[ x^5, x^8], {x, 0, n}]; (* _Michael Somos_, Jun 03 2014 *)
%t A035460 nmax = 100; CoefficientList[Series[Product[1/((1 - x^(8k+3))*(1 - x^(8k+5))), {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 26 2015 *)
%t A035460 nmax = 60; kmax = nmax/8;
%t A035460 s = Flatten[{Range[0, kmax]*8 + 3}~Join~{Range[0, kmax]*8 + 5}];
%t A035460 Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* _Robert Price_, Aug 04 2020 *)
%o A035460 (PARI) a(n)=if(n<0,0,polcoeff(1/prod(k=1, n, 1-(k%8==3||k%8==5)*x^k, 1+x*O(x^n)), n))
%Y A035460 Cf. A035691.
%K A035460 nonn
%O A035460 0,12
%A A035460 _Olivier Gérard_