A237824 - OEIS

A237824

Number of partitions of n such that 2*(least part) >= greatest part.

1, 2, 3, 4, 5, 7, 7, 10, 11, 13, 14, 19, 18, 23, 25, 29, 30, 38, 37, 46, 48, 54, 57, 70, 69, 80, 85, 97, 100, 118, 118, 137, 144, 159, 168, 193, 195, 220, 233, 259, 268, 303, 311, 348, 367, 399, 419, 469, 483, 532, 560, 610, 639, 704, 732, 801, 841, 908, 954

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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

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..300 from John Tyler Rascoe)

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

G.f.: Sum_{k>=1} x^k / Product_{j=k..2*k} (1 - x^j). - Vaclav Kotesovec, Jun 13 2025

a(n) ~ phi^(3/2) * exp(Pi*sqrt(2*n/15)) / (5^(1/4) * sqrt(2*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 14 2025

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 *)

(* or *)

nmax = 100; Rest[CoefficientList[Series[Sum[x^k/Product[1 - x^j, {j, k, 2*k}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 13 2025 *)

(* or *)

nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(2*k - 1))*(1 - x^(2*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax+1)]; s += x^k/(1 - x^k)/p; , {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 14 2025 *)

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

Cf. A237821, A118096, A053263.

The complement is counted by A237820, ranks A362982.

For modes instead of middles we have A362619, counted by A171979.

These partitions have ranks A362981.

A000041 counts integer partitions, strict A000009.

A325347 counts partitions with integer median, complement A307683.