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A334578 Double subfactorials: a(n) = (-1)^floor(n/2) * n!! * Sum_{i=0..floor(n/2)} (-1)^i/(n-2*i)!!.

%I A334578 #41 Nov 27 2020 07:40:15
%S A334578 1,1,1,2,5,11,29,76,233,685,2329,7534,27949,97943,391285,1469144,
%T A334578 6260561,24975449,112690097,474533530,2253801941,9965204131,
%U A334578 49583642701,229199695012,1190007424825,5729992375301,30940193045449,154709794133126,866325405272573
%N A334578 Double subfactorials: a(n) = (-1)^floor(n/2) * n!! * Sum_{i=0..floor(n/2)} (-1)^i/(n-2*i)!!.
%H A334578 Alois P. Heinz, <a href="/A334578/b334578.txt">Table of n, a(n) for n = 0..807</a>
%F A334578 a(n) = n*a(n-2) + (-1)^floor(n/2).
%F A334578 a(2n) = A000354(n).
%F A334578 From _Ryan Brooks_, Oct 25 2020: (Start)
%F A334578 a(2n)/A006882(2n) ~ 1/sqrt(e) = A092605.
%F A334578 a(2n+1)/A006882(2n+1) ~ sqrt(Pi/(2*e))*erfi(1/sqrt(2)) = A306858. (End)
%e A334578 a(5) = (5*3*1)*(1/(1) - 1/(3*1) + 1/(5*3*1)) = 15-5+1 = 11.
%p A334578 a:= proc(n) option remember; `if`(n<2, [0$2, 1$2][n+3],
%p A334578       (n-1)*a(n-2)+(n-2)*a(n-4))
%p A334578     end:
%p A334578 seq(a(n), n=0..32);  # _Alois P. Heinz_, May 06 2020
%t A334578 RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == n a[n-2] + (-1)^Floor[n/2]}, a, {n, 0, 32}] (* _Jean-François Alcover_, Nov 27 2020 *)
%Y A334578 Even bisection gives A000354.
%Y A334578 Cf. A000166, A006882.
%K A334578 nonn
%O A334578 0,4
%A A334578 _Ryan Brooks_, May 06 2020