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A302057 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^5 is zero.

%I A302057 #27 Nov 05 2023 17:42:10
%S A302057 1560,1802,1838,2318,2690,3174,3742,3925,4348,4710,4854,5002,5092,
%T A302057 5210,7484,7615,8796,8846,9500,10345,12110,14178,14972,16203,18010,
%U A302057 19314,20207,20406,20679,24566,25231,27403,27532,28361,31567,31573,35610,35795,37347
%N A302057 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^5 is zero.
%C A302057 Numbers k such that number of partitions of k into an even number of distinct parts equals number of partitions of k into an odd number of distinct parts, with 5 types of each part.
%H A302057 Joerg Arndt, <a href="/A302057/b302057.txt">Table of n, a(n) for n = 1..1212</a> (terms 1..53 from Seiichi Manyama, terms 54..81 from Jean-François Alcover)
%H A302057 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%t A302057 Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[QPochhammer[x]^5, {x, 0, nmax}], x]], 0]]
%t A302057 Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[Sum[(-1)^j x^(j (3 j + 1)/2), {j, -nmax, nmax}]^5, {x, 0, nmax}], x]], 0]]
%t A302057 Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[Exp[-5 Sum[DivisorSigma[1, j] x^j/j, {j, 1, nmax}]], {x, 0, nmax}], x]], 0]]
%t A302057 (* 4th program: *)
%t A302057 sigma[k_] := sigma[k] = DivisorSigma[1, k];
%t A302057 a[0] = 1; a[n_] := a[n] = -5/n Sum[sigma[k] a[n-k], {k, 1, n}];
%t A302057 Reap[For[k = 1, k <= 10^5, k++, If[a[k] == 0, Print[k]; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Dec 20 2018 *)
%o A302057 (PARI) x='x+O('x^30000); v=Vec(eta(x)^5 - 1); for(k=1, #v, if(v[k]==0, print1(k, ", "))); \\ _Altug Alkan_, Mar 31 2018, after _Joerg Arndt_ at A213250
%Y A302057 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m = 1), A213250 (m = 2), A014132 (m = 3), A302056 (m = 4), this sequence (m = 5), A020757 (m = 6), A322043 (m = 15).
%Y A302057 Cf. A000728.
%K A302057 nonn
%O A302057 1,1
%A A302057 _Ilya Gutkovskiy_, Mar 31 2018