A355258 - OEIS

%I #10 Apr 12 2024 14:01:48

%S 0,1,1,5,34,294,3096,38520,553680,9036720,165191040,3344664960,

%T 74321452800,1798531257600,47088252288000,1326311841254400,

%U 39993302622873600,1285497518393088000,43878291581988864000,1585102883250991104000,60420385100090695680000,2423528644964637450240000

%N a(n) = n! * [x^n] (1 - x)*log((1 - x)/(1 - 2*x)).

%F For n>=2, a(n) = (1 + 2^(n-1) * (n-2)) * (n-2)!. - _Vaclav Kotesovec_, Jul 01 2022

%F For n>=2, a(n) = n!*Sum_{k, 0, n - 2} (binomial(n - 2, k)/(k + 2)). - _Detlef Meya_, Apr 12 2024

%p egf := (1 - x)*log((1 - x)/(1 - 2*x)): ser := series(egf, x, 23):

%p seq(n!*coeff(ser, x, n), n = 0..21);

%p # Alternative:

%p a := n -> local k; n! * ifelse(n < 2, n, (2^(n - 1)*(n - 2) + 1) / (n*(n - 1))):

%p seq(a(n), n = 0..21);  # _Peter Luschny_, Apr 12 2024

%t a[0]:=0; a[1]:=1; a[n_]:=n!*Sum[Binomial[n-2,k]/(k+2), {k,0,n-2}];

%t Flatten[Table[a[n],{n,0,21}]] (* _Detlef Meya_, Apr 12 2024 *)

%Y Cf. A355257.

%K nonn

%O 0,4

%A _Peter Luschny_, Jul 01 2022