A363067 Number of partitions p of n such that (1/4)*max(p) is a part of p.
1, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 39, 51, 64, 81, 102, 128, 159, 198, 245, 304, 374, 460, 563, 689, 841, 1023, 1242, 1505, 1819, 2195, 2642, 3173, 3804, 4551, 5435, 6477, 7707, 9151, 10850, 12843, 15175, 17902, 21089, 24802, 29132, 34164, 40012, 46796, 54663, 63766
Offset: 0
Keywords
Examples
a(8) = 3 counts these partitions: 431, 4211, 41111.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Programs
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Mathematica
nmax = 60; CoefficientList[Series[Sum[x^(5*k)/Product[1 - x^j, {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *) nmax = 60; p=1; s=1; Do[p=Expand[p*(1-x^(4*k))*(1-x^(4*k-1))*(1-x^(4*k-2))*(1-x^(4*k-3))]; p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]]; s+=x^(5*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *) -
PARI
a(n) = sum(k=0, n\5, #partitions(n-5*k, 4*k));
Formula
G.f.: Sum_{k>=0} x^(5*k)/Product_{j=1..4*k} (1-x^j).
a(n) ~ Gamma(1/4) * Pi^(1/4) * exp(Pi*sqrt(2*n/3)) / (2^(49/8) * 3^(5/8) * n^(9/8)). - Vaclav Kotesovec, Jun 19 2025