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 A163869 Binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457). 2
 1, 7, 43, 249, 1395, 7653, 41381, 221399, 1175027, 6196725, 32512401, 169863147, 884318973, 4589954619, 23761814955, 122735222505, 632698778835, 3255832730565, 16728131746145, 85826852897675 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also a(n) = sum {i=0..n} binomial(n,n-i) (2*i+1)\$ where i\$ denotes the swinging factorial of i (A056040). REFERENCES Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..300 Peter Luschny, Swinging Factorial. 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 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 *) CROSSREFS Cf. A163842. Sequence in context: A081896 A193656 A125344 * A043553 A049609 A161728 Adjacent sequences:  A163866 A163867 A163868 * A163870 A163871 A163872 KEYWORD nonn AUTHOR Peter Luschny, Aug 06 2009 STATUS approved

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Last modified August 15 09:12 EDT 2018. Contains 313756 sequences. (Running on oeis4.)