A026770 a(n) = T(2n,n), T given by A026769.

1, 2, 7, 28, 120, 538, 2493, 11854, 57558, 284392, 1426038, 7241356, 37173304, 192638992, 1006564439, 5297715628, 28061959428, 149491856978, 800425486692, 4305263668514, 23251846197766, 126044501870378, 685569373724964, 3740339567665558, 20463965229643218, 112250484320225118

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

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

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

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(x + Sqrt(1-4*x) + Sqrt(1-6*x+x^2)) )); // G. C. Greubel, Nov 01 2019

Maple

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

Mathematica

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]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 24 2019 *)

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 */

Sage

def A026770_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 2/(x + sqrt(1-4*x) + sqrt(1-6*x+x^2)) ).list()
A026770_list(30) # G. C. Greubel, Nov 01 2019

Formula

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