A237831 - OEIS

A237831

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

1, 2, 3, 5, 6, 10, 12, 18, 23, 32, 40, 57, 70, 94, 120, 157, 196, 256, 318, 408, 508, 640, 792, 996, 1223, 1518, 1863, 2296, 2798, 3432, 4162, 5070, 6130, 7422, 8936, 10777, 12916, 15500, 18522, 22136, 26348, 31376, 37222, 44160, 52236, 61756, 72824, 85847

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OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..93 from R. J. Mathar)

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

FORMULA

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

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

EXAMPLE

a(6) = 10 counts all the 11 partitions of 6 except 4+1+2.

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

CROSSREFS