login
A327548
Total number of compositions in the compositions of partitions of n.
4
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 16 2019
STATUS
approved