A390116 a(n) is the total number of vertices of degree 3 in Catalan word grid graphs with n parts.

4, 27, 135, 606, 2585, 10725, 43771, 176800, 709308, 2832710, 11277222, 44796870, 177673275, 703921785, 2786742675, 11026630560, 43615236720, 172480297050, 682006922610, 2696608125900, 10662292792242, 42160443165522, 166722867247070, 659372540539776, 2608073656673800

Offset

3,1

Links

Table of n, a(n) for n=3..27.

Aubrey Blecher and Arnold Knopfmacher, Grid graphs of Catalan words, Online Journal of Analytic Combinatorics, Issue 20, 2025. 1-19. See Theorem 5.4 at p. 15.

Formula

G.f.: (2 - 2*sqrt(1 - 4*x) - 3*x - x*sqrt(1 - 4*x) - 7*x^2 + x^2*sqrt(1 - 4*x))/(sqrt(1 - 4*x)*(1 + sqrt(1 - 4*x) - 2*x)).

a(n) ~ 13*4^(n-1)/(2*sqrt(Pi*n)).

Mathematica

CoefficientList[Series[(2-2Sqrt[1-4x]-3x-x*Sqrt[1-4x]-7x^2+x^2*Sqrt[1-4x])/(Sqrt[1-4x](1+Sqrt[1-4x]-2x)), {x, 0, 27}], x]

Crossrefs

Cf. A000108, A000302, A000984, A390111, A390115.

Sequence in context: A395543 A034512 A366486 * A240194 A278358 A274751

Adjacent sequences: A390113 A390114 A390115 * A390117 A390118 A390119

Keyword

nonn

Author

Stefano Spezia, Oct 25 2025

Status

approved