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%I #18 Sep 02 2019 01:53:42
%S 0,0,1,2,8,22,72,218,691,2158,6833,21612,68726,218892,699197,2237450,
%T 7174018,23038582,74097134,238625222,769407486,2483532218,8024499657,
%U 25951580444,83999410292,272098963300,882045339733,2861184745710,9286923094550,30161343633746
%N Next-to-central column of triangle in A059317.
%C Number of h steps in all paths in the first quadrant from (0,0) to (n-1,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because in the 6 (=A128720(3)) paths hhh, hH, Hh, hUD, UhD and UDh we have altogether 8 h-steps. a(n) = Sum_{k=0..n-1} k*A132277(n-1,k). - _Emeric Deutsch_, Sep 03 2007
%C Number of paths in the right half-plane from (0,0) to (n-1,1) consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because we have hhU, HU, hUh, Uhh, UH, DUU, UDU and UUD. Number of h-steps in all paths in the first quadrant from (0,0) to (n-1,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0). Example: a(4)=8 because in the 6 (=A128720(3)) paths from (0,0) to (3,0), namely, hhh, hH, Hh, hUD, UhD and UDh, we have altogether 8 h-steps. a(n) = Sum_{k=0..n-1} k*A132277(n-1,k). - _Emeric Deutsch_, Sep 03 2007
%H W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, <a href="http://www.fq.math.ca/Scanned/35-4/klostermeyer.pdf">A Pascal rhombus</a>, Fibonacci Quarterly, 35 (1997), 318-328.
%H José L. Ramírez, <a href="http://arxiv.org/abs/1511.04577">The Pascal Rhombus and the Generalized Grand Motzkin Paths</a>, arXiv:1511.04577 [math.CO], 2015.
%F G.f.: (1 - z - z^2 - sqrt((1+z-z^2)*(1-3z-z^2)))/(2*sqrt((1+z-z^2)*(1-3z-z^2))). - _Emeric Deutsch_, Sep 03 2007
%F G.f.: (1-z-z^2)/(2*sqrt((1+z-z^2)*(1-3z-z^2))) - 1/2. - _Emeric Deutsch_, Sep 03 2007
%p g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/sqrt((1+z-z^2)*(1-3*z-z^2)): gser:=series(g,z=0,33); seq(coeff(gser,z,n),n=0..29); # _Emeric Deutsch_, Sep 03 2007
%p g:=((1-z-z^2)*1/2)/sqrt((1+z-z^2)*(1-3*z-z^2))-1/2: gser:=series(g,z=0,33): seq(coeff(gser,z,n),n=0..30); # _Emeric Deutsch_, Sep 03 2007
%t t[0, 0] = t[1, 0] = t[1, 1] = t[1, 2] = 1;
%t t[n_ /; n >= 0, k_ /; k >= 0] /; k <= 2n := t[n, k] = t[n-1, k] + t[n-1, k-1] + t[n-1, k-2] + t[n-2, k-2];
%t t[n_, k_] /; n<0 || k<0 || k>2n = 0;
%t a[n_] := t[n-1, n-2];
%t Table[a[n], {n, 0, 29}] (* _Jean-François Alcover_, Aug 07 2018 *)
%Y Cf. A059317, A128720, A132277.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, May 28 2005
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