A238641 Number of partitions p of 2n-1 such that n - (number of parts of p) is a part of p.

0, 0, 1, 4, 9, 21, 37, 69, 113, 187, 286, 449, 657, 976, 1397, 2003, 2788, 3902, 5323, 7284, 9789, 13144, 17405, 23052, 30142, 39379, 50967, 65842, 84368, 107954, 137126, 173893

Offset

1,4

Links

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

Giulio Ruzza and Di Yang, On the spectral problem of the quantum KdV hierarchy, arXiv:2104.01480 [math-ph], 2021.

Example

a(4) counts these partitions of 7:  52, 511, 421, 331.

Mathematica

z = 30; g[n_] := IntegerPartitions[n]; m[p_, t_] := MemberQ[p, t];
Table[Count[g[2 n], p_ /; m[p, n - Length[p]]], {n, z}] (*A238607*)
Table[Count[g[2 n - 1], p_ /; m[p, n - Length[p]]], {n, z}] (*A238641*)
Table[Count[g[2 n + 1], p_ /; m[p, n - Length[p]]], {n, z}] (*A238742*)
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_] := Sum[q[2*n - 1, u, n - u], {u, n - 1}]; Array[a, 100] (* Giovanni Resta, Mar 09 2014 *)

Crossrefs

Cf. A238640, A238742.

Sequence in context: A146948 A033944 A287143 * A038415 A002762 A361152

Adjacent sequences: A238638 A238639 A238640 * A238642 A238643 A238644

Keyword

nonn,easy

Author

Clark Kimberling, Mar 04 2014

Status

approved