A390111 - OEIS

%I #9 Oct 26 2025 10:13:52

%S 4,10,29,91,300,1023,3575,12727,45968,167960,619514,2302990,8617640,

%T 32427585,122611275,465542655,1774086600,6782469540,26004015510,

%U 99953651850,385077767880,1486591659150,5749679124774,22275652390326,86434602692480,335860170462208,1306751215490420

%N a(n) is the total number of vertices of degree 1 in Catalan word grid graphs with n parts.

%H Aubrey Blecher and Arnold Knopfmacher, <a href="https://doi.org/10.61091/ojac20-03">Grid graphs of Catalan words</a>, Online Journal of Analytic Combinatorics, Issue 20, 2025. 1-19. See Theorem 3.1 at p. 9.

%F a(n) = (3*binomial(2*n,n) + 5*binomial(2*n-2, n-1) - binomial(2*n+2, n+1))/2.

%F G.f.: (1 - x - 5*x^2)/(2*x) - (1 - 3*x - 5*x^2)/(2*x*sqrt(1-4*x)).

%F E.g.f.: (exp(2*x)*((1 + 5*x)*BesselI(0, 2*x) - (2 + 5*x)*BesselI(1, 2*x)) - 1 - 5*x)/2.

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

%t a[n_]:=(3Binomial[2n,n]+5Binomial[2n-2,n-1]-Binomial[2n+2,n+1])/2; Array[a,27,2] (* or *)

%t CoefficientList[Series[(1-x-5x^2)/(2x)-(1-3*x-5x^2)/(2x*Sqrt[1-4x]),{x,0,28}],x]

%Y Cf. A000108, A000302, A000984, A390115, A390116.

%K nonn,easy

%O 2,1

%A _Stefano Spezia_, Oct 25 2025