A239956 Number of partitions p of n such that (number of distinct parts of p) = max(p) - min(p).

0, 0, 0, 0, 1, 1, 2, 3, 6, 5, 10, 12, 16, 21, 28, 30, 45, 53, 63, 81, 99, 114, 144, 173, 204, 245, 293, 340, 410, 482, 554, 662, 774, 890, 1044, 1207, 1393, 1619, 1864, 2134, 2464, 2828, 3220, 3701, 4223, 4789, 5474, 6223, 7050, 8004, 9058, 10230, 11579

Offset: 0

Author

Clark Kimberling, Mar 30 2014

Keywords

• nonn
• easy

Examples

			a(8) counts these 6 partitions:  53, 431, 422, 4211, 3311, 311111.
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A239954, A239955, A034296, A239958.

Programs

Mathematica

z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}]  (* A239954 *)
Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (* A239955 *)
Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (* this sequence *)
Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}]  (* A034296 *)
Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (* A239958 *)

PARI

A_x(N) = {my(x='x+O('x^N), g = sum(m=1,N, sum(i=m+2,N, x^(i+m)/((1-x^i)*(1-x^m)) * sum(j=m+1,i-1, (1-x^j)/(x^j) * prod(k=m+1,i-1, (x^k/(1-x^k)))))));
concat([0,0,0,0],Vec(g))}
A_x(51) \\ John Tyler Rascoe, Mar 16 2024

Formula

A239954(n) + a(n) + A034296(n) = A000041(n) for n >= 1.

G.f.: Sum_{m>0} A(x,m), where A(x,m) = Sum_{i>m+1} x^(i+m)/((1-x^i)*(1-x^m)) * Sum_{j=m+1..i-1} ( (1-x^j)/(x^j) * Product_{k=m+1..i-1} (x^k/(1-x^k)) ) is the g.f. for partitions of this kind with min(p) = m. - John Tyler Rascoe, Mar 16 2024