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A327548 Total number of compositions in the compositions of partitions of n.

%I A327548 #14 Dec 17 2020 14:49:44
%S A327548 0,1,4,11,34,85,248,603,1630,4017,10308,24855,63210,150141,369936,
%T A327548 882083,2135606,5023689,12064092,28167919,66828418,155569685,
%U A327548 364983208,844175675,1971322574,4533662817,10498550260,24077361031,55432615194,126492183213,289997946944
%N A327548 Total number of compositions in the compositions of partitions of n.
%H A327548 Alois P. Heinz, <a href="/A327548/b327548.txt">Table of n, a(n) for n = 0..3202</a>
%F A327548 a(n) = Sum_{k=1..n} k * A327549(n,k).
%F A327548 a(n) ~ log(2) * (3/(Pi^2 - 6*log(2)^2))^(1/4) * 2^(n-1) * exp(sqrt((Pi^2 - 6*log(2)^2)*n/3)) / (sqrt(Pi) * n^(1/4)). - _Vaclav Kotesovec_, Sep 19 2019
%e A327548 a(3) = 11 = 1+1+1+1+2+2+3 counts the compositions in 3, 21, 12, 111, 2|1, 11|1, 1|1|1.
%p A327548 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A327548       b(n, i-1)+(p->p+[0, p[1]])(2^(i-1)*b(n-i, min(n-i, i)))))
%p A327548     end:
%p A327548 a:= n-> b(n$2)[2]:
%p A327548 seq(a(n), n=0..32);
%t A327548 b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, b[n, i - 1] + With[{p = 2^(i - 1) b[n - i, Min[n - i, i]]}, p + {0, p[[1]]}]]];
%t A327548 a[n_] := b[n, n][[2]];
%t A327548 a /@ Range[0, 32] (* _Jean-François Alcover_, Dec 17 2020, after _Alois P. Heinz_ *)
%Y A327548 Cf. A000041, A011782, A075900, A326346, A327549.
%K A327548 nonn
%O A327548 0,3
%A A327548 _Alois P. Heinz_, Sep 16 2019