OFFSET
1,2
COMMENTS
By conjugation, also the number of integer partitions of n whose greatest part appears at a middle position, namely at k/2, (k+1)/2, or (k+2)/2 where k is the number of parts. These partitions have ranks A362622. - Gus Wiseman, May 14 2023
LINKS
John Tyler Rascoe, Table of n, a(n) for n = 1..300
FORMULA
G.f.: Sum_{m>0} x^m/(1-x^m) + Sum_{i>0} Sum_{j=1..i} x^((2*i)+j) / Product_{k=0..j} (1 - x^(k+i)). - John Tyler Rascoe, Mar 07 2024
EXAMPLE
a(6) = 7 counts these partitions: 6, 42, 33, 222, 2211, 21111, 111111.
From Gus Wiseman, May 14 2023: (Start)
The a(1) = 1 through a(8) = 10 partitions such that 2*(least part) >= greatest part:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (211) (221) (42) (322) (53)
(1111) (2111) (222) (2221) (332)
(11111) (2211) (22111) (422)
(21111) (211111) (2222)
(111111) (1111111) (22211)
(221111)
(2111111)
(11111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part appears at a middle position:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(11111) (222) (331) (71)
(2211) (2221) (332)
(111111) (1111111) (2222)
(3311)
(22211)
(11111111)
(End)
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}] (* this sequence *)
PROG
(PARI)
N=60; x='x+O('x^N);
gf = sum(m=1, N, (x^m)/(1-x^m)) + sum(i=1, N, sum(j=1, i, x^((2*i)+j)/prod(k=0, j, 1 - x^(k+i))));
Vec(gf) \\ John Tyler Rascoe, Mar 07 2024
CROSSREFS
These partitions have ranks A362981.
KEYWORD
nonn
AUTHOR
Clark Kimberling, Feb 16 2014
STATUS
approved