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%I #23 Jun 07 2023 09:45:54
%S 1,2,6,17,51,154,473,1464,4568,14332,45187,143024,454217,1446604,
%T 4618576,14777451,47371177,152110326,489165277,1575211177,5078690936,
%U 16392526502,52963765321,171282782902,554393341371,1795821017014
%N Row sums of triangle in A059397.
%C 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
%C 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
%H G. C. Greubel, <a href="/A059398/b059398.txt">Table of n, a(n) for n = 0..1000</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
%H W. Klostermeyer et al., <a href="http://www.fq.math.ca/Scanned/35-4/klostermeyer.pdf">A Pascal rhombus</a>, Fibonacci Quarterly, 35 (1976), 318-328.
%F G.f.: (sqrt((1+x-x^2)/(1-3*x-x^2))-1)/x/2. - _Vladeta Jovovic_, Jan 20 2004
%F 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
%p 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
%t 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 *)
%t 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 *)
%o (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 */
%o (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
%o (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
%Y Cf. A128720, A132276.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 29 2001
%E More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001
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