A237820 Number of partitions of n such that 2*(least part) < greatest part.

0, 0, 0, 1, 2, 4, 8, 12, 19, 29, 42, 58, 83, 112, 151, 202, 267, 347, 453, 581, 744, 948, 1198, 1505, 1889, 2356, 2925, 3621, 4465, 5486, 6724, 8212, 9999, 12151, 14715, 17784, 21442, 25795, 30952, 37079, 44315, 52871, 62950, 74827, 88767, 105159, 124335

Offset: 1

Author

Clark Kimberling, Feb 16 2014

Keywords

• nonn
• easy

Examples

			a(6) = 4 counts these partitions:  51, 411, 321, 3111.
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000041, A053263, A118096, A237821, A237824.

Programs

Mathematica

z = 60; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 2 Min[p] < Max[p]], {n, z}]  (* A237820 *)
Table[Count[q[n], p_ /; 2 Min[p] <= Max[p]], {n, z}] (* A237821 *)
Table[Count[q[n], p_ /; 2 Min[p] = = Max[p]], {n, z}](* A118096 *)
Table[Count[q[n], p_ /; 2 Min[p] > Max[p]], {n, z}]  (* A053263 *)
Table[Count[q[n], p_ /; 2 Min[p] >= Max[p]], {n, z}] (* A237824 *)

PARI

A(n) = {concat([0,0,0], Vec(sum(i=1, n, sum(j=1, n-3*i, x^(3*i+j)/prod(k=i, min(n-3*i-j,2*i+j), 1-x^k)))+ O('x^(n+1))))} \\ John Tyler Rascoe, Jun 21 2025

Formula

G.f.: Sum_{i>0} Sum_{j>0} x^(3*i+j) /Product_{k=i..2*i+j} (1 - x^k). - John Tyler Rascoe, Jun 21 2025