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

3,2

Comments

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

Links

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

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

Example

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

Mathematica

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

Prog

(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  */

Crossrefs

Cf. A145879.

Sequence in context: A027650 A172064 A197238 * A026843 A026874 A190866

Adjacent sequences: A182539 A182540 A182541 * A182543 A182544 A182545

Keyword

nonn

Author

N. J. A. Sloane, May 04 2012

Extensions

More terms from Alois P. Heinz, May 30 2012

Status

approved