A163869 Binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).

1, 7, 43, 249, 1395, 7653, 41381, 221399, 1175027, 6196725, 32512401, 169863147, 884318973, 4589954619, 23761814955, 122735222505, 632698778835, 3255832730565, 16728131746145, 85826852897675, 439793834236745, 2251006269442815, 11509340056410735, 58790764269668805

Offset: 0

Author

Peter Luschny, Aug 06 2009

Keywords

• nonn
• changed

Comments

Also a(n) = Sum_{i=0..n} 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. A163842, A163872, A387208.

Programs

Maple

a := proc(n) local i; add(binomial(n,i)/Beta(i+1,i+1), i=0..n) end:

Mathematica

CoefficientList[Series[-Sqrt[x-1]/(5*x-1)^(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[ Binomial[n, n-i]*sf[2*i+1], {i, 0, n}]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jul 26 2013 *)
Table[Hypergeometric2F1[3/2, -n, 1, -4], {n, 0, 20}] (* Vladimir Reshetnikov, Apr 25 2016 *)

Formula

From Vaclav Kotesovec, Oct 21 2012: (Start)

G.f.: -sqrt(x-1)/(5*x-1)^(3/2).

Recurrence: n*a(n) = (6*n+1)*a(n-1) - 5*(n-1)*a(n-2).

a(n) ~ 4*5^(n-1/2)*sqrt(n)/sqrt(Pi).

(End)

a(n) = hypergeom([3/2, -n], [1], -4) = hypergeom([3/2, n+1], [1], 4/5)/(5*sqrt(5)). - Vladimir Reshetnikov, Apr 25 2016

E.g.f.: exp(3*x) * ((1 + 4*x) * BesselI(0,2*x) + 4 * x * BesselI(1,2*x)). - Ilya Gutkovskiy, Nov 19 2021

From Seiichi Manyama, Aug 22 2025: (Start)

a(n) = (1/4)^n * Sum_{k=0..n} 5^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).

a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n,n-k).

a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k). (End)