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%I #24 Mar 17 2020 11:53:49
%S 0,0,0,0,0,0,0,1,2,3,6,9,13,22,33,51,74,104,142,194,261,351,464,616,
%T 802,1047,1344,1716,2186,2766,3473,4367,5448,6774,8375,10329,12685,
%U 15553,18982,23098,28046,33966,40976,49381,59301,71095,85017,101491,120859
%N Number of not unique partition coefficients of n.
%C 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).
%F a(n) = A000041(n) - A070289(n).
%e 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.
%e 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].
%p h := proc(n,k) option remember;
%p if n = 0 then return [1] elif k < 1 then return [] fi;
%p [h(n, k-1)[], seq(map(x -> x*k!^j, h(n-k*j, k-1))[], j=1..n/k)] end:
%p A309897 := proc(n) h(n, n); nops(%) - nops(convert(%, set)) end:
%p seq(A309897(n), n=0..48);
%o (SageMath)
%o def A309897(n):
%o P = Partitions(n)
%o M = set(multinomial(x) for x in P)
%o return P.cardinality() - len(M)
%o [A309897(n) for n in range(29)]
%Y Cf. A000041, A070289.
%K nonn
%O 0,9
%A _Peter Luschny_, Sep 06 2019
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