OFFSET
0,3
COMMENTS
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
LINKS
John Tyler Rascoe, Table of n, a(n) for n = 0..500
FORMULA
G.f.: Sum_{i, j>0} x^(i*j) * Product_{k=1,i-1} ((1-x^(j*k))/(1-x^k)). - John Tyler Rascoe, Apr 03 2024
EXAMPLE
The a(1) = 1 through a(10) = 7 partitions (A = 10):
1 2 3 4 5 6 7 8 9 A
11 111 22 221 33 331 44 333 55
1111 11111 222 2221 332 441 442
111111 1111111 2222 3321 3331
22211 22221 22222
11111111 111111111 222211
1111111111
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Commonest[#]=={Max[#]}&]], {n, 0, 30}]
PROG
(PARI)
A_x(N)={my(x='x+O('x^N), g=sum(i=1, N, sum(j=1, N/i, x^(i*j)*prod(k=1, i-1, (1-x^(j*k))/(1-x^k))))); concat([0], Vec(g))}
A_x(60) \\ John Tyler Rascoe, Apr 03 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 03 2023
STATUS
approved