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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 (list; graph; refs; listen; history; text; internal format)
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).

LINKS

Table of n, a(n) for n=0..48.

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 = [multinomial(x) for x in P]

    return len(M) - len(Set(M))

print [A309897(n) for n in range(49)]

CROSSREFS

Cf. A000041, A070289.

Sequence in context: A308824 A039903 A280422 * A128955 A215523 A306247

Adjacent sequences:  A309894 A309895 A309896 * A309898 A309899 A309900

KEYWORD

nonn

AUTHOR

Peter Luschny, Sep 06 2019

STATUS

approved

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Last modified January 28 13:14 EST 2020. Contains 331321 sequences. (Running on oeis4.)