A364446 Odd bisection of A097514.

1, 2, 17, 205, 3876, 99585, 3313117, 138046940, 6974868139, 419104459913, 29405917751526, 2376498296500063, 218615700758838253, 22667167720595002186, 2626657814273218158997, 337692419653329329932633, 47859496337287704749354668

Offset: 0

Author

Karol A. Penson, Jul 25 2023

Keywords

• nonn

Comments

Closed-form expression in terms of known functions.

a(n) is the number of partitions of a (2n+1)-set without blocks of size 2. - Alois P. Heinz, Jul 25 2023

Crossrefs

Cf. A097514.

Programs

Maple

# Maple program 1:
Digits:=48;
a:= proc(n) round(evalf(sum(p^(2*n + 1)*hypergeom([-n, -n - 1/2],
       [ ], -2/p^2)/p!, p = 1 .. infinity)/exp(1)));
    end:
seq(a(n),n=0..16);
# Alternative formula in terms of generalized Laguerre
# polynomials LaguerreL(n,b,z):
# Maple program 2:
Digits:=48;
a:= proc(n) round(evalf(sum(factor(expand(p^(2*n+1)*n!*
      (-2/p^2)^n*LaguerreL(n,1/2,p^2/2)))/p!,p=1..infinity)/exp(1)));
    end:
seq(a(n),n=0..16);
# third Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(`if`(
       j=2, 0, b(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
a:= n-> b(2*n+1):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2023

Mathematica

b[n_] := b[n] = If[n == 0, 1,
   Sum[If[j == 2, 0, b[n-j]*Binomial[n-1, j-1]], {j, 1, n}]];
a[n_] := b[2n+1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 03 2024, after Alois P. Heinz *)

PARI

my(N=44,x='x+O('x^N)); v=Vec(serlaplace(exp(exp(x)-1-x^2/2))); vector(#v\2,n,v[2*n]) \\ Joerg Arndt, Jul 26 2023

Formula

a(n) = Sum_{p >= 1} (p^(2*n + 1)*hypergeom([-n, -n - 1/2], [ ], -2/p^2)/p!) / exp(1).

a(n) = (2*n+1)! * [x^(2*n+1)] exp(exp(x)-1-x^2/2). - Alois P. Heinz, Jul 25 2023