A097091 Number of partitions of n such that the least part occurs exactly three times.

0, 0, 1, 0, 1, 2, 2, 2, 6, 5, 8, 11, 15, 18, 27, 30, 43, 54, 69, 83, 113, 134, 172, 211, 265, 320, 405, 483, 602, 726, 888, 1064, 1306, 1554, 1884, 2248, 2707, 3213, 3860, 4560, 5446, 6435, 7638, 8990, 10651, 12494, 14734, 17260, 20277, 23683, 27754, 32328

Offset

1,6

Comments

Number of partitions p of n such that 2*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^(3*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) + 3*p(n+3) - p(n+4) - 2*p(n+5) + p(n+6), 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)) + 37*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]], 4]; If[q[[1]] != q[[4]] && q[[2]] == q[[4]], c++]; k++]; c]; Table[ a[n], {n, 52}]
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Length[p] + 2*Min[p]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)
nmax = 60; Rest[CoefficientList[Series[Sum[x^(3*m)/Product[1-x^k, {k, m+1, nmax}], {m, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 04 2025 *)
Table[-PartitionsP[n] + 3 PartitionsP[3 + n] - PartitionsP[4 + n] - 2 PartitionsP[5 + n] + PartitionsP[6 + n], {n, 1, 60}] (* Vaclav Kotesovec, Jul 05 2025 *)

Crossrefs

Cf. A002865, A096373, A097092, A097093.

Sequence in context: A209820 A145890 A382823 * A094204 A088681 A078584

Adjacent sequences: A097088 A097089 A097090 * A097092 A097093 A097094

Keyword

nonn

Author

Robert G. Wilson v, Jul 24 2004

Status

approved