A302056 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^4 is zero.

9, 14, 19, 24, 31, 34, 39, 42, 44, 49, 53, 59, 64, 65, 69, 74, 75, 82, 84, 86, 89, 94, 97, 99, 108, 109, 111, 114, 116, 119, 124, 130, 133, 134, 139, 144, 149, 150, 152, 157, 159, 163, 164, 167, 169, 174, 180, 184, 185, 189, 194, 196, 198, 199, 201, 203, 207, 209

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 4 types of each part.

From Jianing Song, Feb 09 2021: (Start)

The following are equivalent:

- k is in this sequence;

- At least one prime congruent to 5 modulo 6 divides 6*k+1 with an odd exponent;

- 6*k+1 is not of the form x^2 + x*y + y^2, i.e., 6*k+1 is in A034020. (End)

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• 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), this sequence (m = 4), A302057 (m = 5), A020757 (m = 6), A322430 (m = 8), A322431 (m = 10), A322432 (m = 14), A322043 (m = 15), A322433 (m = 26).

Cf. A000727, A034020.

Programs

Mathematica

Flatten[Position[nmax = 210; Rest[CoefficientList[Series[QPochhammer[x]^4, {x, 0, nmax}], x]], 0]]
Flatten[Position[nmax = 210; Rest[CoefficientList[Series[Sum[(-1)^j x^(j (3 j + 1)/2), {j, -nmax, nmax}]^4, {x, 0, nmax}], x]], 0]]
Flatten[Position[nmax = 210; Rest[CoefficientList[Series[Exp[-4 Sum[DivisorSigma[1, j] x^j/j, {j, 1, nmax}]], {x, 0, nmax}], x]], 0]]

PARI

x='x+O('x^999); v=Vec(eta(x)^4 - 1); for(k=1, #v, if(v[k]==0, print1(k, ", "))); \\ Altug Alkan, Mar 31 2018, after Joerg Arndt at A213250