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

1, 2, 5, 16, 53, 206, 817, 3620, 16361, 80218, 401501, 2139512, 11641885, 66599846, 388962953, 2367284236, 14700573137, 94523836850, 619674301621, 4186249123808, 28809504493061, 203556335785342, 1463877667140065, 10777146970619636, 80686484464418233

Offset: 0

Author

Alexander Stoimenow (stoimeno(AT)math.toronto.edu)

Keywords

• nonn

Comments

Binomial transform of { n!/floor(n/2)! }.

Number of symmetric chord diagrams of degree n-1.

Row sums of exponential Riordan array [(1+x), x(1+x)]. - Paul Barry, Apr 17 2007

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..300 (first 200 terms from _Vincenzo Librandi_)
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
• Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233.

Crossrefs

Cf. A047974, A081125.

Programs

Maple

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

Mathematica

a[n_] := Sum[Binomial[n-1, k] k! / Floor[k/2]!, {k, 0, n}];
Array[a, 25] (* Jean-François Alcover, Aug 29 2019 *)
Table[n!*SeriesCoefficient[(1+x)*E^(x+x^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)

Formula

a(n) = A047974(n-1) + (n-1)*A047974(n-2). - Vladeta Jovovic, Aug 06 2006

E.g.f.: (1 + x)*exp(x + x^2). - Vladeta Jovovic, Aug 06 2006

Recurrence: (n-2)*a(n) = (n-3)*a(n-1) + 2*(n-1)^2*a(n-2). - Vaclav Kotesovec, Oct 13 2012

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

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

Extensions

Entry revised by N. J. A. Sloane, Sep 25 2021