A237832 Number of partitions of n such that (greatest part) - (least part) = number of parts.

0, 0, 0, 1, 0, 2, 1, 3, 3, 5, 4, 10, 8, 13, 15, 22, 22, 34, 36, 51, 58, 75, 85, 116, 130, 165, 194, 244, 281, 355, 409, 505, 591, 718, 839, 1022, 1186, 1425, 1668, 1994, 2319, 2765, 3213, 3805, 4429, 5214, 6052, 7119, 8243, 9645, 11169, 13026, 15046, 17511

Offset

1,6

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..96 from R. J. Mathar)

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

Formula

A237830(n) + a(n) + A237833(n) = A000041(n). - R. J. Mathar, Nov 24 2017

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

Example

a(6) = 2 counts these partitions:  4+2, 4+1+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 *)

Prog

(PARI) my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(1/prod(k=1, N, 1-x^k)*1/(1-x)*sum(k=1, N, (-1)^(k-1)*(k*(1-x)*x^(k*(3*k-1)/2)*(1+x^k)-x^(3*k*(k-1)/2+1)*(1-x^(2*k)))))) \\ Seiichi Manyama, May 20 2023

Crossrefs

Cf. A237830, A237831, A237833, A237834.

Sequence in context: A026927 A240863 A288005 * A074500 A107237 A070047

Adjacent sequences: A237829 A237830 A237831 * A237833 A237834 A237835

Keyword

nonn,easy

Author

Clark Kimberling, Feb 16 2014

Status

approved