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A018191 a(n) = Sum_{k=0..n} binomial(n, k) * k! / floor(k/2)!.

%I A018191 #45 Nov 12 2024 09:01:43
%S A018191 1,2,5,16,53,206,817,3620,16361,80218,401501,2139512,11641885,
%T A018191 66599846,388962953,2367284236,14700573137,94523836850,619674301621,
%U A018191 4186249123808,28809504493061,203556335785342,1463877667140065,10777146970619636,80686484464418233
%N A018191 a(n) = Sum_{k=0..n} binomial(n, k) * k! / floor(k/2)!.
%C A018191 Binomial transform of { n!/floor(n/2)! }.
%C A018191 Number of symmetric chord diagrams of degree n-1.
%C A018191 Row sums of exponential Riordan array [(1+x), x(1+x)]. - _Paul Barry_, Apr 17 2007
%H A018191 N. J. A. Sloane, <a href="/A018191/b018191.txt">Table of n, a(n) for n = 0..300</a> (first 200 terms from _Vincenzo Librandi_)
%H A018191 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A018191 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>. [Cached copy]
%H A018191 Alexander Stoimenow, <a href="https://doi.org/10.1016/S0012-365X(99)00347-7">On the number of chord diagrams</a>, Discr. Math. 218 (2000), 209-233.
%F A018191 a(n) = A047974(n-1) + (n-1)*A047974(n-2). - _Vladeta Jovovic_, Aug 06 2006
%F A018191 E.g.f.: (1 + x)*exp(x + x^2). - _Vladeta Jovovic_, Aug 06 2006
%F A018191 Recurrence: (n-2)*a(n) = (n-3)*a(n-1) + 2*(n-1)^2*a(n-2). - _Vaclav Kotesovec_, Oct 13 2012
%F A018191 a(n) ~ 2^(n/2 - 1)*exp(sqrt(n/2) - n/2 - 1/8)*n^(n/2 + 1/2)*(1 + 85/96*sqrt(2)/sqrt(n)). - _Vaclav Kotesovec_, Oct 13 2012
%F A018191 a(n) = -(n-3)*a(n-1) + 3*(n-1)*a(n-2) + 2*(n-1)*(n-2)*a(n-3) for n > 2. - _Seiichi Manyama_, Nov 12 2024
%p A018191 f:=n-> add(binomial(n,k)*k!/floor(k/2)!, k=0..n); [seq(f(n),n=1..40)]; # _N. J. A. Sloane_, Sep 25 2021
%t A018191 a[n_] := Sum[Binomial[n-1, k] k! / Floor[k/2]!, {k, 0, n}];
%t A018191 Array[a, 25] (* _Jean-François Alcover_, Aug 29 2019 *)
%t A018191 Table[n!*SeriesCoefficient[(1+x)*E^(x+x^2),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 13 2012 *)
%Y A018191 Cf. A047974, A081125.
%K A018191 nonn
%O A018191 0,2
%A A018191 Alexander Stoimenow (stoimeno(AT)math.toronto.edu)
%E A018191 Entry revised by _N. J. A. Sloane_, Sep 25 2021