OFFSET
0,2
COMMENTS
Number of paths in the first quadrant from (0,0) to the line x=n, consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) (in other words, left factors of the paths in A128720). Example: a(2)=6 because we have hh, H, UD, hU, Uh and UU. Row sums of triangle in A132276. - Emeric Deutsch, Sep 03 2007
Row sums of the Riordan matrix (g(x),x*g(x)), where g(x) = (1-x-x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^2) (A132276). - Emanuele Munarini, May 05 2011
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
W. Klostermeyer et al., A Pascal rhombus, Fibonacci Quarterly, 35 (1976), 318-328.
FORMULA
G.f.: (sqrt((1+x-x^2)/(1-3*x-x^2))-1)/x/2. - Vladeta Jovovic, Jan 20 2004
a(n) = (1/2)*sum(binomial(2*k,k)*(-1)^(n-k+1)*sum(binomial(i+k-1,i)*binomial(i,n-k-i+1),i=0..n-k+1),k=0..n+1). - Emanuele Munarini, May 05 2011
MAPLE
g:=(1/2)*(sqrt((1+x-x^2)/(1-3*x-x^2))-1)/x: gser:=series(g, x=0, 30): seq(coeff(gser, x, n), n=0..25); # Emeric Deutsch, Sep 03 2007
MATHEMATICA
Table[Sum[Binomial[2k, k](-1)^(n-k+1)Sum[Binomial[i+k-1, i]Binomial[i, n-k-i+1], {i, 0, n-k+1}], {k, 0, n+1}]/2, {n, 0, 28}] (* Emanuele Munarini, May 05 2011 *)
With[{nn = 50}, CoefficientList[Series[(Sqrt[(1 + x - x^2)/(1 - 3*x - x^2)] - 1)/x/2, {x, 0, nn}], x]] (* G. C. Greubel, Jan 29 2018 *)
PROG
(Maxima) makelist(sum(binomial(2*k, k)*(-1)^(n-k+1)*sum(binomial(i+k-1, i)*binomial(i, n-k-i+1), i, 0, n-k+1), k, 0, n+1)/2, n, 0, 28); /* Emanuele Munarini, May 05 2011 */
(PARI) x='x+O('x^30); Vec((sqrt((1+x-x^2)/(1-3*x-x^2))-1)/x/2) \\ G. C. Greubel, Jan 29 2018
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!(Sqrt((1+x-x^2)/(1-3*x-x^2))-1)/(2*x)) // G. C. Greubel, Jan 29 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 29 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001
STATUS
approved