A327550 Number of compositions of partitions of 2n with exactly n compositions.

1, 2, 8, 24, 80, 224, 704, 1920, 5632, 15360, 43008, 114688, 315392, 827392, 2211840, 5767168, 15138816, 38928384, 100925440, 256901120, 657457152, 1660944384, 4202692608, 10527703040, 26424115200, 65699577856, 163477192704, 403995361280, 998043025408

Offset: 0

Author

Alois P. Heinz, Sep 16 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3129

Crossrefs

Cf. A000041, A000079, A327549.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or
      k<1, 0, b(n, i-1, k)+ 2^(i-1)*b(n-i, min(n-i, i), k-1)))
    end:
a:= n-> b(2*n$2, n)-`if`(n=0, 0, b(2*n$2, n-1)):
seq(a(n), n=0..40);

Mathematica

Table[2^n * PartitionsP[n], {n, 0, 30}] (* Vaclav Kotesovec, Oct 02 2020 *)

Formula

a(n) = A327549(2n,n).

a(n) ~ 2^n * exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). - Vaclav Kotesovec, Sep 19 2019

a(n) = A000041(n) * A000079(n). - Alois P. Heinz, Oct 08 2020

G.f.: Product_{k>=1} 1 / (1 - 2^k*x^k). - Ilya Gutkovskiy, Apr 24 2021