A097092 Number of partitions of n such that the least part occurs exactly four times.

0, 0, 0, 1, 0, 1, 1, 3, 2, 4, 5, 9, 9, 14, 16, 26, 29, 40, 48, 67, 79, 105, 126, 165, 196, 253, 303, 385, 459, 572, 687, 852, 1014, 1244, 1482, 1807, 2145, 2595, 3075, 3701, 4375, 5231, 6170, 7350, 8641, 10247, 12025, 14201, 16620, 19557, 22839, 26790, 31209

Offset: 1

Author

Robert G. Wilson v, Jul 24 2004

Keywords

• nonn

Comments

Number of partitions p of n such that 3*min(p) + (number of parts of p) is a part of p. - Clark Kimberling, Feb 28 2014

Links

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

Crossrefs

Cf. A002865, A096373, A097091, A097093.

Programs

Mathematica

a[n_] := Module[{p = IntegerPartitions[n], l = PartitionsP[n], c = 0, k = 1}, While[k < l + 1, q = PadLeft[ p[[k]], 5]; If[ q[[1]] != q[[5]] && q[[2]] == q[[5]], c++ ]; k++ ]; c]; Table[ a[n], {n, 53}]
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Length[p] + 3*Min[p]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)
Table[Count[IntegerPartitions[n],?(Length[Split[#][[-1]]]==4&)],{n,60}] (* _Harvey P. Dale, Jan 18 2021 *)
nmax = 60; Rest[CoefficientList[Series[Sum[x^(4*m)/Product[1-x^k,{k,m+1,nmax}], {m, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 04 2025 *)
Table[-PartitionsP[n] + 4 PartitionsP[4 + n] - PartitionsP[5 + n] - 2 PartitionsP[6 + n] - 2 PartitionsP[7 + n] + PartitionsP[8 + n] + 2 PartitionsP[9 + n] - PartitionsP[10 + n], {n, 1, 60}] (* Vaclav Kotesovec, Jul 05 2025 *)

Formula

G.f.: Sum_{m>0} (x^(4*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) + 4*p(n+4) - p(n+5) - 2*p(n+6) - 2*p(n+7) + p(n+8) + 2*p(n+9) - p(n+10), 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)) + 61*Pi/(24*sqrt(6)))/sqrt(n)). (End)