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A309897
Number of not unique partition coefficients of n.
2
0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 6, 9, 13, 22, 33, 51, 74, 104, 142, 194, 261, 351, 464, 616, 802, 1047, 1344, 1716, 2186, 2766, 3473, 4367, 5448, 6774, 8375, 10329, 12685, 15553, 18982, 23098, 28046, 33966, 40976, 49381, 59301, 71095, 85017, 101491, 120859
OFFSET
0,9
COMMENTS
We call (p1+p2+ ...)! / (p1!*p2!*p3! ...) a 'partition coefficient' of n if (p1, p2, p3, ...) is a partition, n = p1 + p2 + ... and denote it by P(n, p).
FORMULA
a(n) = A000041(n) - A070289(n).
EXAMPLE
a(7) = 1 because the partition coefficients of 7 are [1, 7, 21, 42, 35, 105, 210, 140, 210, 420, 840, 630, 1260, 2520, 5040], P(7, [3, 2, 2]) = P(7, [4, 1, 1, 1]) = 210 and all other partition coefficients are unique.
We say that two partitions of n are multinomial-equivalent if they have the same partition coefficient. For instance [6, 2, 2, 1, 1] ~ [5, 4, 1, 1, 1] ~ [5, 3, 2, 2] and [6, 4, 1, 1, 1, 1, 1] ~ [6, 3, 2, 2, 1, 1] ~ [5, 4, 3, 1, 1, 1] ~ [5, 3, 3, 2, 2].
MAPLE
h := proc(n, k) option remember;
if n = 0 then return [1] elif k < 1 then return [] fi;
[h(n, k-1)[], seq(map(x -> x*k!^j, h(n-k*j, k-1))[], j=1..n/k)] end:
A309897 := proc(n) h(n, n); nops(%) - nops(convert(%, set)) end:
seq(A309897(n), n=0..48);
PROG
(SageMath)
def A309897(n):
P = Partitions(n)
M = set(multinomial(x) for x in P)
return P.cardinality() - len(M)
[A309897(n) for n in range(29)]
CROSSREFS
Sequence in context: A357619 A360362 A280422 * A128955 A215523 A306247
KEYWORD
nonn
AUTHOR
Peter Luschny, Sep 06 2019
STATUS
approved