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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 301 terms from John Tyler Rascoe)

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

Example

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

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}]

Prog

(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

Crossrefs

Cf. A241090, A241132.

Sequence in context: A188381 A005576 A339592 * A339397 A129373 A139078

Adjacent sequences: A241128 A241129 A241130 * A241132 A241133 A241134

Keyword

nonn,easy

Author

Clark Kimberling, Apr 24 2014

Status

approved