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

1,9

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

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)

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 *)

Crossrefs

Cf. A002865, A096373, A097091, A097092.

Sequence in context: A344680 A330147 A241037 * A056877 A267232 A269606

Adjacent sequences: A097090 A097091 A097092 * A097094 A097095 A097096

Keyword

nonn

Author

Robert G. Wilson v, Jul 24 2004

Status

approved