A238629 Number of partitions p of n such that n - 2*(number of parts of p) is a part of p.

0, 0, 0, 0, 1, 1, 4, 4, 9, 9, 18, 18, 31, 31, 51, 51, 79, 79, 119, 119, 173, 173, 248, 248, 347, 347, 480, 480, 654, 654, 883, 883, 1178, 1178, 1561, 1561, 2049, 2049, 2674, 2674, 3464, 3464, 4464, 4464, 5717, 5717, 7290, 7290, 9246, 9246, 11680, 11680

Offset

1,7

Links

Giovanni Resta, Table of n, a(n) for n = 1..1000

Example

a(7) counts these partitions:  511, 43, 421, 331.

Mathematica

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, n - 2*Length[p]]], {n, 50}]
p[n_, k_] := p[n, k] = If[k == 1 || n == k, 1, If[k > n, 0, p[n - 1, k - 1] + p[n - k, k]]]; q[n_, k_, e_] := q[n, k, e] = If[n - e < k - 1 , 0, If[k == 1, If[n == e, 1, 0], p[n - e, k - 1]]]; a[n_] := a[n] = Sum[q[n, u, n - 2*u], {u, (n - 1)/2}]; Array[a, 100] (* Giovanni Resta, Mar 09 2014 *)

Crossrefs

Cf. A000027 = (number of partitions p of n such that n - (number of parts of p) is a part of p) = n-2 for n >=3.

Sequence in context: A008794 A075709 A332777 * A192032 A116682 A168157

Adjacent sequences: A238626 A238627 A238628 * A238630 A238631 A238632

Keyword

nonn,easy

Author

Clark Kimberling, Mar 02 2014

Status

approved