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

1560, 1802, 1838, 2318, 2690, 3174, 3742, 3925, 4348, 4710, 4854, 5002, 5092, 5210, 7484, 7615, 8796, 8846, 9500, 10345, 12110, 14178, 14972, 16203, 18010, 19314, 20207, 20406, 20679, 24566, 25231, 27403, 27532, 28361, 31567, 31573, 35610, 35795, 37347

Offset: 1

Author

Ilya Gutkovskiy, Mar 31 2018

Keywords

• nonn

Comments

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.

Links

• Joerg Arndt, Table of n, a(n) for n = 1..1212 (terms 1..53 from Seiichi Manyama, terms 54..81 from Jean-François Alcover)
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Crossrefs

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).

Cf. A000728.

Programs

Mathematica

Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[QPochhammer[x]^5, {x, 0, nmax}], x]], 0]]
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]]
Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[Exp[-5 Sum[DivisorSigma[1, j] x^j/j, {j, 1, nmax}]], {x, 0, nmax}], x]], 0]]
(* 4th program: *)
sigma[k_] := sigma[k] = DivisorSigma[1, k];
a[0] = 1; a[n_] := a[n] = -5/n Sum[sigma[k] a[n-k], {k, 1, n}];
Reap[For[k = 1, k <= 10^5, k++, If[a[k] == 0, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 20 2018 *)

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