A104625 -id:A104625 - cp's OEIS frontend

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

Clark Kimberling

nonn

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

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

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

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

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

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[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 *)

{ 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 */

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

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

A369627 Expansion of 1/(1 - x^2/(1-9*x)^(1/3)).

Original entry on oeis.org

1, 0, 1, 3, 19, 132, 991, 7740, 62020, 505857, 4180132, 34889514, 293518072, 2485191753, 21153817090, 180865139538, 1552289627872, 13366436688402, 115425148203235, 999256943147094, 8670047414816233, 75375298322580081, 656465004512563546

Offset: 0

Seiichi Manyama, Feb 06 2024

nonn

Vaclav Kotesovec, Graph - the asymptotic ratio (300000 terms)

Cf. A362206, A369940.

Cf. A104625.

CoefficientList[Series[1/(1 - x^2/(1-9*x)^(1/3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 19 2024 *)
Flatten[{{1, 0, 1, 3, 19, 132}, RecurrenceTable[{9 (-12 + n) (-19 + 3 n) (-14 + 3 n) a[-8 + n] - 6 (-628 + 368 n - 63 n^2 + 3 n^3) a[-7 + n] + (-13 + n) (-4 + n) (-2 + n) a[-6 + n] + 81 (-12 + n) (-19 + 3 n) (-14 + 3 n) a[-3 + n] - 9 (-6960 + 3662 n - 585 n^2 + 27 n^3) a[-2 + n] + 3 (-14 + 3 n) (112 - 47 n + 3 n^2) a[-1 + n] - (-13 + n) (-4 + n) (-2 + n) a[n] == 0, a[6] == 991, a[7] == 7740, a[8] == 62020, a[9] == 505857, a[10] == 4180132, a[11] == 34889514, a[12] == 293518072, a[13] == 2485191753}, a, {n, 6, 20}]}] (* Vaclav Kotesovec, Feb 19 2024 *)

my(N=30, x='x+O('x^N)); Vec(1/(1-x^2/(1-9*x)^(1/3)))

a(n) = Sum_{k=0..floor(n/2)} 9^(n-2*k) * binomial(n-1-5*k/3,n-2*k).

a(n) ~ (r-9)^(4/3) * r^(5/3) * r^n / (2*r-15), where r = 9.0000169349284790514638157821699098461789951085871459872133... = is the largest real root of the equation r^5*(r-9) = 1. - Vaclav Kotesovec, Feb 19 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula