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A326346 Total number of partitions in the partitions of compositions of n.

%I A326346 #37 Dec 05 2020 06:41:52
%S A326346 0,1,4,14,47,151,474,1457,4414,13210,39155,115120,336183,976070,
%T A326346 2819785,8110657,23239662,66362960,188930728,536407146,1519205230,
%U A326346 4293061640,12106883585,34079016842,95762829405,268670620736,752676269695,2105751165046,5883798478398
%N A326346 Total number of partitions in the partitions of compositions of n.
%H A326346 Alois P. Heinz, <a href="/A326346/b326346.txt">Table of n, a(n) for n = 0..2313</a>
%F A326346 a(n) = Sum_{k=1..n} k * A060642(n,k).
%F A326346 a(n) ~ c * d^n * n, where d = A246828 = 2.69832910647421123126399866618837633... and c = 0.171490233695958246364725709205670983251448838158816... - _Vaclav Kotesovec_, Sep 14 2019
%e A326346 a(3) = 14 = 1+1+1+2+2+2+2+3 counts the partitions in 3, 21, 111, 2|1, 11|1, 1|2, 1|11, 1|1|1.
%p A326346 b:= proc(n) option remember; `if`(n=0, [1, 0], (p-> p+
%p A326346       [0, p[1]])(add(combinat[numbpart](j)*b(n-j), j=1..n)))
%p A326346     end:
%p A326346 a:= n-> b(n)[2]:
%p A326346 seq(a(n), n=0..32);
%t A326346 b[n_] := b[n] = If[n==0, {1, 0}, Function[p, p + {0, p[[1]]}][Sum[ PartitionsP[j] b[n-j], {j, 1, n}]]];
%t A326346 a[n_] := b[n][[2]];
%t A326346 a /@ Range[0, 32] (* _Jean-François Alcover_, Dec 05 2020, after _Alois P. Heinz_ *)
%Y A326346 Cf. A000041, A030267, A055887, A060642, A304969, A325915, A327548.
%K A326346 nonn
%O A326346 0,3
%A A326346 _Alois P. Heinz_, Sep 11 2019