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A211867 a(n) = A097609(2*n-1,n), n>0; a(0)=1.

%I A211867 #39 Apr 27 2024 04:00:11
%S A211867 1,0,2,3,18,50,215,735,2898,10668,41202,156090,601623,2308878,8923343,
%T A211867 34487453,133749330,519277512,2020262660,7869597840,30699524018,
%U A211867 119894389380,468768069882,1834589752182,7186572436887,28175111736300,110547143014050,434049816801900
%N A211867 a(n) = A097609(2*n-1,n), n>0; a(0)=1.
%H A211867 Michael De Vlieger, <a href="/A211867/b211867.txt">Table of n, a(n) for n = 0..1665</a>
%H A211867 D. Kruchinin and V. Kruchinin, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL18/Kruchinin/kruch9.html">A Generating Function for the Diagonal T2n,n in Triangles</a>, Journal of Integer Sequence,  Vol. 18 (2015), article 15.4.6.
%H A211867 Eric Marberg, <a href="https://arxiv.org/abs/1709.07446">On some actions of the 0-Hecke monoids of affine symmetric groups</a>, arXiv:1709.07996 [math.CO], 2017.
%F A211867 G.f.: x*G'(x)/G(x), where G(x) is the g.f. of A055113.
%F A211867 G.f.: x * d/dx (log(sqrt(12*x+2*sqrt(1-4*x)+2)/4-sqrt(1-4*x)/4-1/4)).
%F A211867 a(n) = sum(j=0..n, C(2*j+n-1,j)*(-1)^(n+j)*C(2*n,n-j))/2, n>0; a(0)=1.
%F A211867 a(n) = A097609(2*n-1,n), n>0; a(0)=1. (Corrected by _M. F. Hasler_, Feb 12 2013)
%F A211867 a(n) = Sum_{j=0..n/2} (binomial(2*n,j)*binomial(n-j-1,n-2*j))/2. - _Vladimir Kruchinin_, Oct 05 2015
%F A211867 a(n) ~ 2^(2*n-1) / sqrt(5*Pi*n). - _Vaclav Kotesovec_, Apr 27 2024
%p A211867 a := n -> (-1)^n*binomial(2*n-1,n-1)*hypergeom([-n,n/2,(n+1)/2], [n,n+1], 4):
%p A211867 seq(simplify(a(n)), n=0..27); # _Peter Luschny_, Nov 02 2016
%t A211867 a[n_] := ((-1)^(3*n)*(2*n)!*HypergeometricPFQ[{(n+1)/2, -n, n/2}, {n, n+1}, 4])/(2*n!^2); a[0]=1; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Feb 13 2013, from A097609 *)
%o A211867 (PARI) a(n) = if(n==0, 1, sum(k=0, n/2, (binomial(2*n,k)*binomial(n-k-1,n-2*k))/2)); \\ _Altug Alkan_, Oct 05 2015
%K A211867 nonn
%O A211867 0,3
%A A211867 _Vladimir Kruchinin_, Feb 12 2013