OFFSET
0,2
COMMENTS
Also a(n) = sum {i=0..n} (-1)^(n-i) binomial(n,n-i) (2*i+1)$ where i$ denotes the swinging factorial of i (A056040).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..300
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
FORMULA
O.g.f.: A(x)=1/(1-x*M(x))^3, M(x) - o.g.f. of A001006. a(n) = sum(k^3/n *sum(C(n,j)*C(j,2*j-n-k), j=0..n), k=1..n). - Vladimir Kruchinin, Sep 06 2010
Recurrence: n*a(n) = (2*n+3)*a(n-1) + 3*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 21 2012
a(n) ~ 4*3^(n-1/2)*sqrt(n)/sqrt(Pi). - Vaclav Kotesovec, Oct 21 2012
a(n) = (-1)^n*hypergeom([-n,3/2], [1], 4). - Peter Luschny, Apr 26 2016
From Seiichi Manyama, Aug 22 2025: (Start)
a(n) = (-1/4)^n * Sum_{k=0..n} (-3)^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
a(n) = Sum_{k=0..n} (-1)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 3^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k). (End)
MAPLE
a := proc(n) local i; add((-1)^(n-i)*binomial(n, i)/Beta(i+1, i+1), i=0..n) end:
seq(simplify((-1)^n*hypergeom([-n, 3/2], [1], 4)), n=0..26); # Peter Luschny, Apr 26 2016
MATHEMATICA
CoefficientList[Series[Sqrt[x+1]/(1-3*x)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 21 2012 *)
sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Sum[(-1)^(n-i)*Binomial[n, n-i]*sf[2*i+1], {i, 0, n}]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jul 26 2013 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Aug 06 2009
STATUS
approved