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

0,2

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

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

D-finite with recurrence 2*n*(132*n-445)*(n+2)*(n+1)*a(n) -n*(n+1)*(5587*n^2-23082*n+12800)*a(n-1) +2*n*(n-1)*(22870*n^2-114505*n+116854)*a(n-2) +2*(-90081*n^4+818062*n^3-2626791*n^2+3517598*n-1622544)*a(n-3) +4*(85519*n^4-1071535*n^3+4986308*n^2-10177616*n+7647024)*a(n-4) +(-269235*n^4+4490125*n^3-27985152*n^2+77217236*n-79534224)*a(n-5) +4*(2*n-11)*(8203*n^3-117312*n^2+557264*n-879984)*a(n-6) -4*(n-6)*(307*n-1414)*(2*n-11)*(2*n-13)*a(n-7)=0. - R. J. Mathar, Jun 30 2026

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 *)

Prog

(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 */
(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
(SageMath)
def A026770_list(prec):
    P.<x> = 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

Crossrefs

Cf. A000108, A006318, A026781 (Invert transform), A104625, A109980.

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

Sequence in context: A150656 A150657 A150658 * A241371 A368970 A010683

Adjacent sequences: A026767 A026768 A026769 * A026771 A026772 A026773

Keyword

nonn

Author

Clark Kimberling

Status

approved