A341071 Number of compositions of 2n into n squarefree parts.

1, 1, 3, 7, 23, 76, 267, 932, 3263, 11410, 40028, 140955, 498467, 1769249, 6299282, 22485487, 80435919, 288275461, 1034862558, 3720499175, 13393720428, 48275699654, 174194970601, 629187941167, 2274710068067, 8230764250326, 29805342037721, 108009638665648

Offset: 0

Author

Alois P. Heinz, Feb 04 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A005117, A333047, A338464, A338465, A341072.

Programs

Maple

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..35);

Mathematica

b[n_, t_] := b[n, t] =
  If[n == 0, If[t == 0, 1, 0], If[t < 1, 0, Sum[
  If[SquareFreeQ[j], b[n - j, t - 1], 0], {j, 1, n}]]];
a[n_] := b[2n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 26 2023, from Maple program *)

Formula

a(n) ~ c * d^n / sqrt(n), where d = 3.6921332109291502908930783402282652076342401140592914... and c = 0.27220098720735952890181498005903942865018992276116... - Vaclav Kotesovec, Feb 14 2021