A344262 a(0)=1; for n>0, a(n) = a(n-1)*n+1 if n is even, (a(n-1)+1)*n otherwise.

1, 2, 5, 18, 73, 370, 2221, 15554, 124433, 1119906, 11199061, 123189682, 1478276185, 19217590418, 269046265853, 4035693987810, 64571103804961, 1097708764684354, 19758757764318373, 375416397522049106, 7508327950440982121, 157674886959260624562

Offset: 0

Author

Amrit Awasthi, May 13 2021

Keywords

• nonn

Examples

			a(0) = 1;
a(1) = (a(0)+1)*1 =  (1+1)*1 =   2;
a(2) = (a(1)*2)+1 =  (2*2)+1 =   5;
a(3) = (a(2)+1)*3 =  (5+1)*3 =  18;
a(4) = (a(3)*4)+1 = (18*4)+1 =  73;
a(5) = (a(4)+1)*5 = (73+1)*5 = 370.
		

Crossrefs

Cf. A033540, A073743, A090805, A137204, A155521, A344317.

Programs

Maple

a:= proc(n) a(n):= n*a(n-1) + n^(n mod 2) end: a(0):= 1:
seq(a(n), n=0..22);  # Alois P. Heinz, May 14 2021

Mathematica

a[1] = 1; a[n_] := a[n] = If[OddQ[n], (n - 1)*a[n - 1] + 1, (n - 1)*(a[n - 1] + 1)]; Array[a, 25] (* Amiram Eldar, May 13 2021 *)

Formula

E.g.f.: (x+1)*cosh(x)/(1-x). - Alois P. Heinz, May 14 2021

Lim_{n->infinity} a(n)/n! = 2*cosh(1) = A137204 = 2*A073743. - Amrit Awasthi, May 15 2021

a(n) = A344317(n) - A155521(n-1) for n > 0. - Alois P. Heinz, May 18 2021