A237830 Number of partitions of n such that (greatest part) - (least part) < number of parts.

1, 2, 3, 4, 6, 8, 11, 15, 20, 27, 36, 47, 62, 81, 105, 135, 174, 222, 282, 357, 450, 565, 707, 880, 1093, 1353, 1669, 2052, 2517, 3077, 3753, 4565, 5539, 6704, 8097, 9755, 11730, 14075, 16854, 20142, 24029, 28611, 34009, 40355, 47807, 56542, 66772, 78728

Offset: 1

Author

Clark Kimberling, Feb 16 2014

Keywords

• nonn
• easy

Examples

			a(6) = 8 counts these partitions:  6, 3+3, 4+1+1, 3+2+1, 2+2+2, 3+1+1+1, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..95 from R. J. Mathar)
• George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.

Crossrefs

Cf. A000070, A237831-A237834.

Programs

Mathematica

z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)

PARI

my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*x/(1-x)*sum(k=1, N, (-1)^(k-1)*x^(3*k*(k-1)/2)*(1-x^(2*k)))) \\ Seiichi Manyama, May 20 2023

Formula

a(n) + A237834(n) = A000041(n). - R. J. Mathar, Nov 24 2017

G.f.: (1/Product_{k>=1} (1-x^k)) * (x/(1-x)) * Sum_{k>=1} (-1)^(k-1) * x^(3*k*(k-1)/2) * (1-x^(2*k)). (See Andrews' preprint.) - Seiichi Manyama, May 20 2023