A097093 Number of partitions of n such that the least part occurs exactly five times.

0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 4, 4, 8, 9, 14, 16, 23, 27, 39, 48, 62, 76, 100, 120, 159, 190, 241, 292, 367, 443, 552, 663, 816, 980, 1200, 1430, 1742, 2075, 2504, 2979, 3575, 4232, 5063, 5980, 7114, 8382, 9930, 11663, 13773, 16140, 18980, 22190, 26017

Offset: 1

Author

Robert G. Wilson v, Jul 24 2004

Keywords

• nonn

Comments

In general, if k>=1 and g.f. = Sum_{m>0} (x^(k*m) / Product_{i>m} (1-x^i)), then a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3^(3/2)/(Pi*sqrt(2)) + (24*k - 35)*Pi/(24*sqrt(6)))/sqrt(n)). - Vaclav Kotesovec, Jul 05 2025

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002865, A096373, A097091, A097092.

Programs

Mathematica

f[n_] := Block[{p = IntegerPartitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[ p[[k]], 6]; If[ q[[1]] != q[[6]] && q[[2]] == q[[6]], c++ ]; k++ ]; c]; Table[ f[n], {n, 54}]
Table[Count[IntegerPartitions[n],?(Length[Split[#][[-1]]]==5&)],{n,60}] (* _Harvey P. Dale, Feb 07 2022 *)
nmax = 60; Rest[CoefficientList[Series[Sum[x^(5*m)/Product[1-x^k,{k,m+1,nmax}], {m, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 04 2025 *)
Table[-PartitionsP[n] + 5 PartitionsP[5 + n] - PartitionsP[6 + n] - 2 PartitionsP[7 + n] - 2 PartitionsP[8 + n] - 3 PartitionsP[9 + n] + 3 PartitionsP[10 + n] + PartitionsP[11 + n] + 2 PartitionsP[12 + n] - PartitionsP[13 + n] - 2 PartitionsP[14 + n] + PartitionsP[15 + n], {n, 1, 60}] (* Vaclav Kotesovec, Jul 05 2025 *)

Formula

G.f.: Sum_{m>0} (x^(5*m) / Product_{i>m} (1-x^i)). More generally, g.f. for number of partitions of n such that the least part occurs exactly k times is Sum_{m>0} (x^(k*m) / Product_{i>m} (1-x^i)). Vladeta Jovovic

From Vaclav Kotesovec, Jul 05 2025: (Start)

a(n) = -p(n) + 5*p(n+5) - p(n+6) - 2*p(n+7) - 2*p(n+8) - 3*p(n+9) + 3*p(n+10) + p(n+11) + 2*p(n+12) - p(n+13) - 2*p(n+14) + p(n+15), where p(n) = A000041(n).

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3^(3/2)/(Pi*sqrt(2)) + 85*Pi/(24*sqrt(6)))/sqrt(n)). (End)