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A364446 Odd bisection of A097514.

%I A364446 #26 May 03 2024 09:29:21
%S A364446 1,2,17,205,3876,99585,3313117,138046940,6974868139,419104459913,
%T A364446 29405917751526,2376498296500063,218615700758838253,
%U A364446 22667167720595002186,2626657814273218158997,337692419653329329932633,47859496337287704749354668
%N A364446 Odd bisection of A097514.
%C A364446 Closed-form expression in terms of known functions.
%C A364446 a(n) is the number of partitions of a (2n+1)-set without blocks of size 2. - _Alois P. Heinz_, Jul 25 2023
%F A364446 a(n) = Sum_{p >= 1} (p^(2*n + 1)*hypergeom([-n, -n - 1/2], [ ], -2/p^2)/p!) / exp(1).
%F A364446 a(n) = (2*n+1)! * [x^(2*n+1)] exp(exp(x)-1-x^2/2). - _Alois P. Heinz_, Jul 25 2023
%p A364446 # Maple program 1:
%p A364446 Digits:=48;
%p A364446 a:= proc(n) round(evalf(sum(p^(2*n + 1)*hypergeom([-n, -n - 1/2],
%p A364446        [ ], -2/p^2)/p!, p = 1 .. infinity)/exp(1)));
%p A364446     end:
%p A364446 seq(a(n),n=0..16);
%p A364446 # Alternative formula in terms of generalized Laguerre
%p A364446 # polynomials LaguerreL(n,b,z):
%p A364446 # Maple program 2:
%p A364446 Digits:=48;
%p A364446 a:= proc(n) round(evalf(sum(factor(expand(p^(2*n+1)*n!*
%p A364446       (-2/p^2)^n*LaguerreL(n,1/2,p^2/2)))/p!,p=1..infinity)/exp(1)));
%p A364446     end:
%p A364446 seq(a(n),n=0..16);
%p A364446 # third Maple program:
%p A364446 b:= proc(n) option remember; `if`(n=0, 1, add(`if`(
%p A364446        j=2, 0, b(n-j)*binomial(n-1, j-1)), j=1..n))
%p A364446     end:
%p A364446 a:= n-> b(2*n+1):
%p A364446 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 25 2023
%t A364446 b[n_] := b[n] = If[n == 0, 1,
%t A364446    Sum[If[j == 2, 0, b[n-j]*Binomial[n-1, j-1]], {j, 1, n}]];
%t A364446 a[n_] := b[2n+1];
%t A364446 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 03 2024, after _Alois P. Heinz_ *)
%o A364446 (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
%Y A364446 Cf. A097514.
%K A364446 nonn
%O A364446 0,2
%A A364446 _Karol A. Penson_, Jul 25 2023