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
Keywords
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.
Programs
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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
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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