A237834 Number of partitions of n such that (greatest part) - (least part) >= number of parts.

0, 0, 0, 1, 1, 3, 4, 7, 10, 15, 20, 30, 39, 54, 71, 96, 123, 163, 208, 270, 342, 437, 548, 695, 865, 1083, 1341, 1666, 2048, 2527, 3089, 3784, 4604, 5606, 6786, 8222, 9907, 11940, 14331, 17196, 20554, 24563, 29252, 34820, 41327, 49016, 57982, 68545, 80833

Offset

1,6

Links

R. J. Mathar, Table of n, a(n) for n = 1..95

George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.

Formula

A237830(n)+a(n) = A000041(n). - R. J. Mathar, Nov 24 2017

Example

a(7) = 4 counts these partitions:  6+1, 5+2, 5+1+1, 4+2+1.

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 *)
Table[Count[IntegerPartitions[n], _?(#[[1]]-#[[-1]]>=Length[#]&)], {n, 50}] (* Harvey P. Dale, Jul 21 2023 *)

Crossrefs

Cf. A237830, A237831, A237832, A237833.

Sequence in context: A050572 A249668 A105343 * A147955 A147789 A047625

Adjacent sequences: A237831 A237832 A237833 * A237835 A237836 A237837

Keyword

nonn,easy

Author

Clark Kimberling, Feb 16 2014

Status

approved