A363075 Number of partitions of n such that 3*(least part) + 1 = greatest part.
0, 0, 0, 0, 1, 1, 2, 3, 6, 6, 10, 12, 18, 20, 27, 32, 42, 47, 59, 67, 85, 94, 113, 126, 152, 169, 198, 220, 257, 282, 326, 359, 413, 452, 512, 563, 639, 695, 781, 853, 958, 1041, 1161, 1261, 1402, 1524, 1685, 1827, 2021, 2186, 2407, 2604, 2861, 3088, 3385, 3657, 4002, 4316, 4704, 5069, 5531
Offset: 1
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..5000
Programs
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Mathematica
nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(3*k - 2))*(1 - x^(3*k - 1))*(1 - x^(3*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax + 1)]; s += x^(4*k + 1)/(1 - x^k)/(1 - x^(3*k + 1))/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 19 2025 *) -
PARI
my(N=70, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(4*k+1)/prod(j=k, 3*k+1, 1-x^j))))
Formula
G.f.: Sum_{k>=1} x^(4*k+1)/Product_{j=k..3*k+1} (1-x^j).
a(n) ~ c * A376815^sqrt(n) / sqrt(n), where c = 0.33761... - Vaclav Kotesovec, Jun 20 2025