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

0, 1, 4, 11, 34, 85, 248, 603, 1630, 4017, 10308, 24855, 63210, 150141, 369936, 882083, 2135606, 5023689, 12064092, 28167919, 66828418, 155569685, 364983208, 844175675, 1971322574, 4533662817, 10498550260, 24077361031, 55432615194, 126492183213, 289997946944

Offset: 0

Author

Alois P. Heinz, Sep 16 2019

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3202

Crossrefs

Cf. A000041, A011782, A075900, A326346, A327549.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      b(n, i-1)+(p->p+[0, p[1]])(2^(i-1)*b(n-i, min(n-i, i)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..32);

Mathematica

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]]}]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 32] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A327549(n,k).

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