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

1, 0, 2, 3, 18, 50, 215, 735, 2898, 10668, 41202, 156090, 601623, 2308878, 8923343, 34487453, 133749330, 519277512, 2020262660, 7869597840, 30699524018, 119894389380, 468768069882, 1834589752182, 7186572436887, 28175111736300, 110547143014050, 434049816801900

Offset: 0

Author

Vladimir Kruchinin, Feb 12 2013

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1665
• D. Kruchinin and V. Kruchinin, A Generating Function for the Diagonal T2n,n in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.
• Eric Marberg, On some actions of the 0-Hecke monoids of affine symmetric groups, arXiv:1709.07996 [math.CO], 2017.

Programs

Maple

a := n -> (-1)^n*binomial(2*n-1,n-1)*hypergeom([-n,n/2,(n+1)/2], [n,n+1], 4):
seq(simplify(a(n)), n=0..27); # Peter Luschny, Nov 02 2016

Mathematica

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 *)

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

Formula

G.f.: x*G'(x)/G(x), where G(x) is the g.f. of A055113.

G.f.: x * d/dx (log(sqrt(12*x+2*sqrt(1-4*x)+2)/4-sqrt(1-4*x)/4-1/4)).

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.

a(n) = A097609(2*n-1,n), n>0; a(0)=1. (Corrected by M. F. Hasler, Feb 12 2013)

a(n) = Sum_{j=0..n/2} (binomial(2*n,j)*binomial(n-j-1,n-2*j))/2. - Vladimir Kruchinin, Oct 05 2015

a(n) ~ 2^(2*n-1) / sqrt(5*Pi*n). - Vaclav Kotesovec, Apr 27 2024