A026940 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A026300.

1, 6, 38, 256, 1805, 13162, 98469, 751656, 5831451, 45847770, 364498596, 2925337352, 23668977163, 192859753310, 1581188102590, 13034447714688, 107971181472779, 898274382703314, 7502546644142842, 62884859093960160, 528788663216036559, 4459599092506030110

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Tewodros Amdeberhan, Moa Apagodu, and Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015.

Crossrefs

Cf. A002026, A026300.

Programs

Maple

a := n -> n*hypergeom([1/2 - n, 1 - n], [3], 4);
seq(simplify(a(n)), n = 1..22); # Peter Luschny, Sep 05 2022

Mathematica

a[n_] := n Hypergeometric2F1[1/2-n, 1-n, 3, 4]; Array[a, 22] (* Jean-François Alcover, Sep 22 2018 *)

PARI

a(n) = sum(k=0, n, binomial(2*n, 2*k+1)*binomial(2*k+1, k)/(k+2)); \\ Michel Marcus, Jul 29 2015

Formula

a(n) = Sum_{k=0..n} binomial(2*n, 2*k+1)*binomial(2*k+1, k)/(k+2), see Amdeberhan link. - Michel Marcus, Jul 29 2015

a(n) = n*hypergeom([1/2 - n, 1 - n], [3], 4). - Jean-François Alcover, Sep 22 2018

a(n) = A002026(2*n)/2. - Mark van Hoeij, Sep 05 2022

Extensions

More terms from Michel Marcus, Jul 29 2015