A026770 - OEIS

%I #24 Sep 08 2022 08:44:49

%S 1,2,7,28,120,538,2493,11854,57558,284392,1426038,7241356,37173304,

%T 192638992,1006564439,5297715628,28061959428,149491856978,

%U 800425486692,4305263668514,23251846197766,126044501870378,685569373724964,3740339567665558,20463965229643218,112250484320225118

%N a(n) = T(2n,n), T given by A026769.

%C Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when below the diagonal, (1,1). - _Alois P. Heinz_, Sep 14 2016

%H Alois P. Heinz, <a href="/A026770/b026770.txt">Table of n, a(n) for n = 0..1000</a>

%F O.g.f.: 1/(1-x*(C(x)+S(x))), where C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108 and S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318. - _Max Alekseyev_, Dec 02 2015

%p seq(coeff(series(2/(x + sqrt(1-4*x) + sqrt(1-6*x+x^2)), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Nov 01 2019

%t T[n_, k_] := T[n, k] = Which[k==0 || k==n, 1, n==2 && k==1, 2, k<=(n-1)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], True, T[n-1, k-1] + T[n-1, k]];

%t a[n_] := T[2n, n];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 24 2019 *)

%o (PARI) { C = (1-sqrt(1-4*x+O(x^51)))/2/x; S = (1-x-sqrt(1-6*x+x^2 +O(x^51)))/2/x; Vec(1/(1-x*(C+S))) } /* _Max Alekseyev_, Dec 02 2015 */

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(x + Sqrt(1-4*x) + Sqrt(1-6*x+x^2)) )); // _G. C. Greubel_, Nov 01 2019

%o (Sage)

%o def A026770_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( 2/(x + sqrt(1-4*x) + sqrt(1-6*x+x^2)) ).list()

%o A026770_list(30) # _G. C. Greubel_, Nov 01 2019

%Y Cf. A000108, A006318, A026781, A104625, A109980.

%Y Cf. A026769, A026771, A026772, A026773, A026774, A026775, A026776, A026777, A026778, A026779.

%K nonn

%O 0,2

%A _Clark Kimberling_