cp's OEIS Frontend

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

%I A026940 #35 Aug 06 2024 00:03:00
%S A026940 1,6,38,256,1805,13162,98469,751656,5831451,45847770,364498596,
%T A026940 2925337352,23668977163,192859753310,1581188102590,13034447714688,
%U A026940 107971181472779,898274382703314,7502546644142842,62884859093960160,528788663216036559,4459599092506030110
%N A026940 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A026300.
%H A026940 Tewodros Amdeberhan, Moa Apagodu, and Doron Zeilberger, <a href="http://arxiv.org/abs/1507.07660">Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle</a>, arXiv:1507.07660 [math.CO], 2015.
%F A026940 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
%F A026940 a(n) = n*hypergeom([1/2 - n, 1 - n], [3], 4). - _Jean-François Alcover_, Sep 22 2018
%F A026940 a(n) = A002026(2*n)/2. - _Mark van Hoeij_, Sep 05 2022
%p A026940 a := n -> n*hypergeom([1/2 - n, 1 - n], [3], 4);
%p A026940 seq(simplify(a(n)), n = 1..22); # _Peter Luschny_, Sep 05 2022
%t A026940 a[n_] := n Hypergeometric2F1[1/2-n, 1-n, 3, 4]; Array[a, 22] (* _Jean-François Alcover_, Sep 22 2018 *)
%o A026940 (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
%Y A026940 Cf. A002026, A026300.
%K A026940 nonn
%O A026940 1,2
%A A026940 _Clark Kimberling_
%E A026940 More terms from _Michel Marcus_, Jul 29 2015