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

Author

Clark Kimberling, Mar 04 2014

Keywords

• nonn
• easy

Examples

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

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.

Crossrefs

Cf. A238640, A238742.

Programs

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 *)