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

%I A163872 #26 Aug 22 2025 11:03:22
%S A163872 1,5,19,67,227,751,2445,7869,25107,79567,250793,786985,2460397,
%T A163872 7667921,23832931,73902627,228692115,706407903,2178511449,6708684009,
%U A163872 20632428249,63380014845,194486530791,596213956023,1826103432573,5588435470401,17089296473655
%N A163872 Inverse binomial transform of the beta numbers 1/beta(n+1,n+1) (A002457).
%C A163872 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).
%H A163872 Vincenzo Librandi, <a href="/A163872/b163872.txt">Table of n, a(n) for n = 0..300</a>
%H A163872 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163872 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%F A163872 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
%F A163872 Recurrence: n*a(n) = (2*n+3)*a(n-1) + 3*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 21 2012
%F A163872 a(n) ~ 4*3^(n-1/2)*sqrt(n)/sqrt(Pi). - _Vaclav Kotesovec_, Oct 21 2012
%F A163872 a(n) = (-1)^n*hypergeom([-n,3/2], [1], 4). - _Peter Luschny_, Apr 26 2016
%F A163872 From _Seiichi Manyama_, Aug 22 2025: (Start)
%F A163872 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)).
%F A163872 a(n) = Sum_{k=0..n} (-1)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
%F A163872 a(n) = Sum_{k=0..n} (-1)^k * 3^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k). (End)
%p A163872 a := proc(n) local i; add((-1)^(n-i)*binomial(n,i)/Beta(i+1,i+1),i=0..n) end:
%p A163872 seq(simplify((-1)^n*hypergeom([-n,3/2], [1], 4)),n=0..26); # _Peter Luschny_, Apr 26 2016
%t A163872 CoefficientList[Series[Sqrt[x+1]/(1-3*x)^(3/2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 21 2012 *)
%t A163872 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 *)
%Y A163872 Cf. A163772.
%K A163872 nonn,changed
%O A163872 0,2
%A A163872 _Peter Luschny_, Aug 06 2009