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A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.

%I A305106 #11 Dec 14 2020 07:49:58
%S A305106 1,1,2,4,7,12,21,34,55,87,138,211,324,486,727,1079,1584,2305,3337,
%T A305106 4789,6830,9712,13689,19225,26841,37322,51598,71108,97580,133350,
%U A305106 181558,246335,332991,448706,602607,806732,1077333,1433885,1903682,2520246,3328549,4383929
%N A305106 Number of unitary factorizations of Heinz numbers of integer partitions of n. Number of multiset partitions of integer partitions of n with pairwise disjoint blocks.
%H A305106 Alois P. Heinz, <a href="/A305106/b305106.txt">Table of n, a(n) for n = 0..1000</a>
%e A305106 The a(6) = 21 unitary factorizations:
%e A305106 (13) (21) (22) (25) (27) (28) (30) (36) (40) (48) (64)
%e A305106 (2*11) (2*15) (3*7) (3*10) (3*16) (4*7) (4*9) (5*6) (5*8)
%e A305106 (2*3*5)
%e A305106 The a(6) = 21 multiset partitions:
%e A305106 {{6}}
%e A305106 {{2,4}}
%e A305106 {{1,5}}
%e A305106 {{3,3}}
%e A305106 {{2,2,2}}
%e A305106 {{1,1,4}}
%e A305106 {{1,2,3}}
%e A305106 {{1,1,2,2}}
%e A305106 {{1,1,1,3}}
%e A305106 {{1,1,1,1,2}}
%e A305106 {{1,1,1,1,1,1}}
%e A305106 {{1},{5}}
%e A305106 {{1},{2,3}}
%e A305106 {{2},{4}}
%e A305106 {{2},{1,3}}
%e A305106 {{2},{1,1,1,1}}
%e A305106 {{1,1},{4}}
%e A305106 {{1,1},{2,2}}
%e A305106 {{3},{1,2}}
%e A305106 {{3},{1,1,1}}
%e A305106 {{1},{2},{3}}
%t A305106 Table[Sum[BellB[Length[Union[y]]],{y,IntegerPartitions[n]}],{n,30}]
%t A305106 (* Second program: *)
%t A305106 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = n - i j}, b[t, Min[t, i - 1], k]], {j, 1, n/i}] k + b[n, i - 1, k]]];
%t A305106 T[n_, k_] := Sum[b[n, n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]/k!;
%t A305106 a[n_] := Sum[T[n, k], {k, 0, Floor[(Sqrt[1 + 8n] - 1)/2]}];
%t A305106 a /@ Range[0, 50] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ in A321878 *)
%Y A305106 Cf. A000110, A001055, A001221, A001970, A034444, A089233, A258466, A259936, A281116, A285572, A305078, A305079.
%Y A305106 Row sums of A321878.
%K A305106 nonn
%O A305106 0,3
%A A305106 _Gus Wiseman_, May 25 2018