A237826 Number of partitions of n such that 4*(least part) = greatest part.
0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 9, 12, 16, 20, 26, 31, 38, 47, 55, 67, 78, 92, 106, 126, 145, 167, 190, 219, 247, 288, 320, 366, 410, 466, 520, 591, 654, 739, 820, 924, 1018, 1148, 1263, 1415, 1562, 1740, 1911, 2136, 2342, 2607, 2859, 3169, 3469, 3849, 4208
Offset: 1
Examples
a(8) = 3 counts these partitions: 431, 4211, 41111.
Links
- David A. Corneth, Table of n, a(n) for n = 1..10000
Programs
-
Mathematica
z = 64; q[n_] := q[n] = IntegerPartitions[n]; Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}] (* A237825*) Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}] (* A237826 *) Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}] (* A237827 *) Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *) Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *) Table[Count[IntegerPartitions[n],?(#[[1]]==4#[[-1]]&)],{n,60}] (* _Harvey P. Dale, Jun 15 2023 *) kmax = 55; Sum[x^(5k)/Product[1 - x^j, {j, k, 4 k}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, May 30 2024, after Seiichi Manyama *) -
PARI
my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=k, 4*k, 1-x^j)))) \\ Seiichi Manyama, May 14 2023
Formula
G.f.: Sum_{k>=1} x^(5*k)/Product_{j=k..4*k} (1-x^j). - Seiichi Manyama, May 14 2023
a(n) ~ c * d^sqrt(n) / sqrt(n), where d = 4.9219345... and c = 0.1699648... - Vaclav Kotesovec, Jun 19 2025