A025567 a(n) = T(n,n+1), where T is the array defined in A025564.

1, 4, 13, 40, 120, 356, 1050, 3088, 9069, 26620, 78133, 229384, 673699, 1979628, 5820195, 17121312, 50394579, 148413996, 437324919, 1289330520, 3803175474, 11223840012, 33139076292, 97889042384, 289276841475, 855205791076, 2529279459099

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1000
• Jean-Luc Baril, Richard Genestier, Sergey Kirgizov, Pattern distributions in Dyck paths with a first return decomposition constrained by height, arXiv:1911.03119 [math.CO], 2019.
• Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv preprint arXiv:1203.6792 [math.CO], 2012 and J. Int. Seq. 17 (2014) #14.1.5

Crossrefs

Pairwise sums of A014531.

Programs

Mathematica

T[, 0] = 1; T[1, 1] = 2; T[n, k_] /; 0 <= k <= 2n := T[n, k] = T[n-1, k-2] + T[n-1, k-1] + T[n-1, k]; T[, ] = 0;
a[n_] := T[n+1, n+3];
Array[a, 27] (* Jean-François Alcover, Oct 30 2018 *)

PARI

x='x+O('x^66); Vec((x^2-1-sqrt(1+x)*(x^2+2*x-1)/sqrt(1-3*x))/(2*x^3)) \\ Joerg Arndt, May 01 2013

Formula

G.f.: (x^2-1-sqrt(1+x)*(x^2+2*x-1)/sqrt(1-3*x))/(2*x^3). - Mark van Hoeij, May 01 2013

Conjecture: (n+3)*a(n) +4*(-n-2)*a(n-1) +2*a(n-2) +8*(n-1)*a(n-3) +3*(n-3)*a(n-4)=0. - R. J. Mathar, Apr 03 2015

Conjecture: (n-1)*(n-2)*(n+3)*a(n) -2*n*(n-2)*(n+2)*a(n-1) -3*n*(n-1)^2*a(n-2)=0. - R. J. Mathar, Apr 03 2015

a(n) ~ 2 * 3^(n + 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, May 02 2024