OFFSET
0,3
COMMENTS
Top left terms of powers of the production matrix M generates sequence A102403. - Gary W. Adamson, Jan 30 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric S. Egge, Kailee Rubin, Snow Leopard Permutations and Their Even and Odd Threads, arXiv:1508.05310 [math.CO], 2015
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
FORMULA
G.f.: f(x) satisfies x*f(x)^3 - (x+1)*f(x)^2 + (2*x+1)*f(x) - x = 0 . - Eric Rowland, Mar 29 2013
The Sapounakis et al. reference gives an explicit formula.
From Gary W. Adamson, Jan 30 2012: (Start)a(n) is the sum of top row terms in M^(n-1), where M = the following infinite square production matrix:
1, 1, 0, 0, 0, 0, ...
0, 1, 1, 0, 0, 0, ...
1, 0, 1, 1, 0, 0, ...
1, 1, 0, 1, 1, 0, ...
1, 1, 1, 0, 1, 1, ... (End)
a(n) ~ sqrt(8 + 5*sqrt(2) + sqrt(2*(11 + 8*sqrt(2))/7))/4 * ((1 + sqrt(13 + 16*sqrt(2)))/2)^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 27 2015
EXAMPLE
a(6) = 63 since the top row of M^5 = (17, 17, 13, 10, 5, 1), sum of terms = 63.
MAPLE
A135306 := proc(n, k) if n =0 then 1 ; else add((-1)^(j-k)*binomial(n-k, j-k)*binomial(2*n-3*j, n-j+1), j=k..floor((n-1)/2)) ; %*binomial(n, k)/n ; fi ; end: A135307 := proc(n) A135306(n, 0) ; end: for n from 0 to 30 do printf("%a, ", A135307(n)) ; od: # R. J. Mathar, Dec 08 2007
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1$2, 2, 4][n+1],
(2*n*(n-1)*(28*n^2-56*n-3)*a(n-1)
+(140*n^4-630*n^3+1063*n^2-699*n+144)*a(n-2)
-12*(n-3)*(14*n^3-42*n^2+16*n+21)*a(n-3)
+23*(n-3)*(n-4)*(28*n^2-14*n-3)*a(n-4))/
(n*(n+1)*(28*n^2-70*n+39)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Sep 13 2014
MATHEMATICA
a[n_] := Sum[(-1)^j*Binomial[n, j]*Binomial[2*n-3*j, n-j+1], {j, 0, (n-1)/2}]/n; a[0] = 1; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Nov 27 2014, after R. J. Mathar *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 07 2007
EXTENSIONS
More terms from R. J. Mathar, Dec 08 2007
STATUS
approved