A163872 Inverse binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

1, 5, 19, 67, 227, 751, 2445, 7869, 25107, 79567, 250793, 786985, 2460397, 7667921, 23832931, 73902627, 228692115, 706407903, 2178511449, 6708684009, 20632428249, 63380014845, 194486530791, 596213956023, 1826103432573, 5588435470401, 17089296473655

Offset: 0

Author

Peter Luschny, Aug 06 2009

Keywords

• nonn
• changed

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.

Crossrefs

Cf. A163772.

Programs

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 *)

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)