A241131 Number of partitions p of n such that (maximal multiplicity over the parts of p) = number of 1s in p.

0, 1, 1, 2, 3, 4, 7, 9, 13, 18, 26, 32, 47, 60, 79, 104, 137, 173, 227, 285, 365, 461, 583, 724, 912, 1129, 1403, 1729, 2137, 2611, 3211, 3906, 4765, 5777, 7010, 8450, 10213, 12263, 14738, 17637, 21113, 25158, 30008, 35638, 42333, 50130, 59346, 70035, 82663

Offset: 0

Author

Clark Kimberling, Apr 24 2014

Keywords

• nonn
• easy

Examples

			a(6) counts these 7 partitions:  51, 411, 321, 3111, 2211, 21111, 111111.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..9500 (terms 0..300 from John Tyler Rascoe, terms 301..2000 from Alois P. Heinz)

Crossrefs

Cf. A241090, A241132.

Programs

Maple

b:= proc(n,i,m) option remember; `if`(i=1, `if`(n>=m, 1, 0),
      add(b(n-i*j, i-1, max(j, m)), j=0..n/i))
    end:
a:= n-> `if`(n=0, 0, b(n$2, 0)):
seq(a(n), n=0..48);  # Alois P. Heinz, Mar 15 2024

Mathematica

z = 30; m[p_] := Max[Map[Length, Split[p]]]; Table[Count[IntegerPartitions[n], p_ /; m[p] == Count[p, 1]], {n, 0, z}]

PARI

A_x(N)={my(x='x+O('x^N),g=sum(i=1, N, x^i*prod(j=2, N, (1-x^(j*(i+1)))/(1-x^j))));
concat([0],Vec(g))}
A_x(50) \\ John Tyler Rascoe, Mar 12 2024

Formula

G.f.: Sum_{i>0} x^i * Product_{j>1} ((1 - x^(j*(i+1)))/(1 - x^j)). - John Tyler Rascoe, Mar 12 2024

a(n) ~ c * exp(Pi*sqrt(2*n/3)) / n, where c = 0.07449179... - Vaclav Kotesovec, Jun 21 2025