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

1,8

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

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)

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 *)

Crossrefs

Cf. A002865, A096373, A097091, A097093.

Sequence in context: A001612 A275901 A305369 * A241417 A211363 A376901

Adjacent sequences: A097089 A097090 A097091 * A097093 A097094 A097095

Keyword

nonn

Author

Robert G. Wilson v, Jul 24 2004

Status

approved