A182542 Second diagonal of triangle in A145879.

1, 8, 46, 232, 1093, 4944, 21778, 94184, 401930, 1698160, 7119516, 29666704, 123012781, 508019104, 2091005866, 8582372584, 35141476126, 143595498544, 585720020356, 2385430111024, 9701814930466, 39411044641888, 159926316674356, 648348726966672, 2626193752638388

Offset: 3

Author

N. J. A. Sloane, May 04 2012

Keywords

• nonn

Comments

Sum of valley heights over all Dyck n-paths. - David Scambler, Oct 05 2012

Examples

			Dyck 4-paths with nonzero valley heights are: UUUD(2)UDDD, UUUDD(1)UDD, UUD(1)UUDDD, UUD(1)UD(1)UDD, UUD(1)UDD(0)UD, and UD(0)UUD(1)UDD, with valley heights shown in parentheses, giving a(4) = 8. - _David Scambler_, Oct 05 2012
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..200

Crossrefs

Cf. A145879.

Programs

Mathematica

a[n_] := 4^(n - 1) - n CatalanNumber[n];
Array[a, 25, 3] (* Peter Luschny, Jun 08 2020 *)

Maxima

a(n):=2*sum((4^i*binomial(2*(n-i),n-i-2))/(n-i),i,0,n-1); /* Vladimir Kruchinin, Mar 29 2019  */

Formula

G.f. appears to be (1-2*x-sqrt(1-4*x))^2/(4*x*(1-4*x)). - Mark van Hoeij, Apr 19 2013

a(n) ~ 2^(2*n-2) * (1-4/(sqrt(Pi*n))). - Vaclav Kotesovec, Aug 13 2013

a(n) = 2*Sum_{i=0..n-1} 4^i*C(2*(n-i),n-i-2)/(n-i). - Vladimir Kruchinin, Mar 29 2019

a(n) = 4^(n-1) - C(2*n,n)*n/(n+1). - Vladimir Kruchinin, Jun 08 2020

Extensions

More terms from Alois P. Heinz, May 30 2012