A238742 - OEIS

%I #7 Mar 12 2014 10:25:40

%S 0,0,1,5,13,31,59,109,180,301,461,712,1051,1547,2200,3138,4349,6036,

%T 8211,11146,14901,19908,26232,34513,44953,58412,75244,96752,123448,

%U 157201,198931,251155

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

%H Giovanni Resta, <a href="/A238742/b238742.txt">Table of n, a(n) for n = 1..1000</a>

%e a(4) counts these partitions of 9:  72, 711, 621, 531, 441.

%t z = 30; g[n_] := IntegerPartitions[n]; m[p_, t_] := MemberQ[p, t];

%t Table[Count[g[2 n], p_ /; m[p, n - Length[p]]], {n, z}] (*A238607*)

%t Table[Count[g[2 n - 1], p_ /; m[p, n - Length[p]]], {n, z}] (*A238641*)

%t Table[Count[g[2 n + 1], p_ /; m[p, n - Length[p]]], {n, z}] (*A238742*)

%t 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 12 2014 *)

%Y Cf. A238640, A238741.

%K nonn,easy

%O 1,4

%A _Clark Kimberling_, Mar 04 2014