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Number of non-isomorphic multiset partitions of weight n where each part has a different length.
8

%I #8 Feb 08 2020 20:41:55

%S 1,1,2,7,12,35,111,247,624,1843,6717,15020,46847,124808,412577,

%T 1658973,4217546,12997734,40786810,126971940,437063393,2106317043,

%U 5499108365,19037901867,59939925812,210338815573,683526043801,2741350650705,14848209030691,41533835240731,151548411269815

%N Number of non-isomorphic multiset partitions of weight n where each part has a different length.

%C The number of non-isomorphic multiset partitions of weight n is A007716(n).

%H Andrew Howroyd, <a href="/A326026/b326026.txt">Table of n, a(n) for n = 0..50</a>

%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions:

%e {{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}

%e {{1,2}} {{1,2,2}} {{1,1,2,2}}

%e {{1,2,3}} {{1,2,2,2}}

%e {{1},{1,1}} {{1,2,3,3}}

%e {{1},{2,2}} {{1,2,3,4}}

%e {{1},{2,3}} {{1},{1,1,1}}

%e {{2},{1,2}} {{1},{1,2,2}}

%e {{1},{2,2,2}}

%e {{1},{2,3,3}}

%e {{1},{2,3,4}}

%e {{2},{1,2,2}}

%e {{3},{1,2,3}}

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o D(p,n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); polcoef(prod(k=1, #u, 1 + u[k]*x^k + O(x*x^n)), n)/prod(i=1, #v, i^v[i]*v[i]!)}

%o a(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ _Andrew Howroyd_, Feb 08 2020

%Y Row sums of A332260.

%Y Cf. A007716, A007837, A303546, A306017, A316980, A316983, A319560, A326514, A326517, A326533.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 13 2019

%E Terms a(11) and beyond from _Andrew Howroyd_, Feb 08 2020