A059398 Row sums of triangle in A059397.

1, 2, 6, 17, 51, 154, 473, 1464, 4568, 14332, 45187, 143024, 454217, 1446604, 4618576, 14777451, 47371177, 152110326, 489165277, 1575211177, 5078690936, 16392526502, 52963765321, 171282782902, 554393341371, 1795821017014

Offset: 0

Author

N. J. A. Sloane, Jan 29 2001

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A128720, A132276.

Programs

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!(Sqrt((1+x-x^2)/(1-3*x-x^2))-1)/(2*x)) // G. C. Greubel, Jan 29 2018

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 *)

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

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

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001