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A363224
Number of integer compositions of n in which the least part appears more than once.
6
0, 1, 1, 5, 8, 21, 44, 94, 197, 416, 857, 1766, 3621, 7392, 15032, 30493, 61708, 124646, 251359, 506203, 1018279, 2046454, 4109534, 8246985, 16540791, 33160051, 66451484, 133122753, 266612828, 533839069, 1068701695, 2139110054, 4281063708, 8566862025
OFFSET
1,4
COMMENTS
Also the number of multisets of length n covering an initial interval of positive integers with more than one co-mode.
LINKS
FORMULA
G.f.: Sum_{i>0} (x^(2*i) * (x-1)^3)/((x^i + x - 1)*(x^(i+1) + x - 1)^2). - John Tyler Rascoe, Jul 06 2024
EXAMPLE
The a(1) = 0 through a(6) = 21 compositions:
. (11) (111) (22) (113) (33)
(112) (131) (114)
(121) (311) (141)
(211) (1112) (222)
(1111) (1121) (411)
(1211) (1113)
(2111) (1122)
(11111) (1131)
(1212)
(1221)
(1311)
(2112)
(2121)
(2211)
(3111)
(11112)
(11121)
(11211)
(12111)
(21111)
(111111)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Count[#, Min@@#]>1&]], {n, 15}]
PROG
(PARI)
C_x(N)={my(x='x+O('x^N), h=sum(i=1, N, (x^(2*i)*(x-1)^3)/((x^i+x-1)*(x^(i+1)+x-1)^2))); concat([0], Vec(h))}
C_x(35) \\ John Tyler Rascoe, Jul 06 2024
CROSSREFS
The complement is counted by A105039.
For partitions instead of compositions we have A117989.
Row sums of columns k > 1 of A238342.
If all parts appear more than once we have A240085, for partitions A007690.
If the least part appears exactly twice we have A241862.
For greatest instead of least we have A363262, see triangle A238341.
A000041 counts integer partitions, strict A000009.
A032020 counts strict compositions.
A067029 gives last exponent in prime factorization, first A071178.
A261982 counts compositions with some part appearing more than once.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
Sequence in context: A294124 A120036 A036381 * A277369 A140419 A292851
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 04 2023
STATUS
approved