A050147 a(n) = T(n,n-1), array T as in A050143. Also T(2n+1,n), array T as in A055807.

1, 3, 12, 56, 280, 1452, 7700, 41456, 225648, 1238420, 6840988, 37986984, 211842696, 1185635388, 6655993380, 37463920608, 211350457824, 1194706644516, 6765300359468, 38370431711000, 217931108199672

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Crossrefs

Cf. A002003, A006318, A050143, A055807.

Programs

Mathematica

a[n_]:=Binomial[2*n-3,n-1]*Hypergeometric2F1[-n+1,-n,-2*n+3,-1];
Table[a[n],{n,1,21}] (* Detlef Meya, Dec 04 2023 *)

Maxima

a(n):=if n=1 then 1 else sum((binomial(n,i+1))*binomial(n+i-2,n-2),i,0,n-2)+binomial(2*n-3,n-2); /* Vladimir Kruchinin, Nov 25 2014 */

Formula

From Vladimir Kruchinin, Nov 25 2014: (Start)

G.f.: x*((-x^2 + 4*x + 1)/(2*sqrt(x^2 - 6*x + 1)) -x/2 + 1/2).

For n >= 2, a(n) = C(2*n-3,n-2) + Sum_{i=0..n-2} C(n,i+1)*C(n+i-2,n-2). (End)

a(n) ~ (1 + sqrt(2))^(2*n-2) / (2^(1/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 14 2021

a(n) = binomial(2*n-3, n-1)*hypergeom([-n+1, -n], [-2*n+3], -1). - Detlef Meya, Dec 04 2023

D-finite with recurrence (-n+1)*a(n) +(2*n+3)*a(n-1) +3*(8*n-29)*a(n-2) +(-10*n+49)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Jul 31 2025