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

0,3

Links

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A127154 A062460 A098324 * A144791 A180305 A060925

Adjacent sequences: A327545 A327546 A327547 * A327549 A327550 A327551

Keyword

nonn

Author

Alois P. Heinz, Sep 16 2019

Status

approved