A059714 -id:A059714 - cp's OEIS frontend

A026737 a(n) = T(2*n,n), T given by A026736.

1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, 108628550, 456710589, 1922354351, 8098984433, 34147706833, 144068881455, 608151037123, 2568318694867, 10850577045131, 45856273670841

Offset: 0

Clark Kimberling

nonn

Is this the same sequence as A111279? - Andrew S. Plewe, May 09 2007

Yes, see the Callan reference "A bijection...". - Joerg Arndt, Feb 29 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Callan, A bijection for two sequences in OEIS, arXiv:1602.08347 [math.CO], 2016.

Cf. A026736, A111279, A059714.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-5*x+4*x^2 -(1-5*x)*Sqrt(1-4*x))/(2*x*(1-4*x-x^2)) )); // G. C. Greubel, Jul 16 2019

T[, 0]=T[n, n_]=1; T[n_, k_]:= T[n, k]= If[EvenQ[n] && k==(n-2)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 22 2018 *)
CoefficientList[Series[(1-5x+4x^2 -(1-5x)*Sqrt[1-4x])/(2*x*(1-4x-x^2)), {x, 0, 30}], x] (* G. C. Greubel, Jul 16 2019 *)

my(x='x+O('x^30)); Vec((1-5*x+4*x^2 -(1-5*x)*sqrt(1-4*x))/(2*x*(1-4*x-x^2))) \\ G. C. Greubel, Jul 16 2019

@CachedFunction
def T(n, k):
    if (k==0 or k==n): return 1
    elif (mod(n,2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
    else: return T(n-1, k-1) + T(n-1, k)
[T(2*n, n) for n in (0..30)] # G. C. Greubel, Jul 16 2019

G.f.: (1-5*x+4*x^2 -(1-5*x)*sqrt(1-4*x))/(2*x*(1-4*x-x^2)). - G. C. Greubel, Jul 16 2019
a(n) ~ (47 - 21*sqrt(5)) * (2 + sqrt(5))^(n+2) / (2*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019
G.f. G satisfies 0 = G^2*(x^3 + 4*x^2 - x) + G*(4*x^2 - 5*x + 1) + 4*x - 1. - F. Chapoton, Oct 16 2021

A059712 Number of stacked directed animals on the square lattice.

Original entry on oeis.org

1, 2, 6, 19, 63, 213, 729, 2513, 8703, 30232, 105236, 366849, 1280131, 4470354, 15619386, 54595869, 190891131, 667590414, 2335121082, 8168950665, 28580354769, 100000811433, 349918126509, 1224476796543, 4285005630969

Offset: 1

Mireille Bousquet-Mélou, Feb 08 2001

nonn

The generating function is simply derived from the generating function for directed animals. A triangular lattice version exists.

			x + 2*x^2 + 6*x^3 + 19*x^4 + 63*x^5 + 213*x^6 + 729*x^7 + ...

M. Bousquet-Mélou and A. Rechnitzer, Lattice animals and heaps of dimers
M. Bousquet-Mélou and A. Rechnitzer, Lattice animals and heaps of dimers, Discrete Math. 258 (2002), no. 1-3, 235-274.
Florian Schager and Michael Wallner, A Bijection between Stacked Directed Polyominoes and Motzkin Paths with Alternative Catastrophes, arXiv:2406.16417 [math.CO], 2024. See p. 5.

Directed animals: A005773.

Cf. A001519, A059714.

gf := ((1-2*x)*(1-3*x)-(1-4*x)*sqrt((1-3*x)*(1+x)))/(2*x*(2-7*x)): s := series(gf, x, 50): for i from 1 to 100 do printf(`%d,`,coeff(s,x,i)) od:

CoefficientList[ ((1-2*x)*(1-3*x)-(1-4*x)*Sqrt[(1-3*x)*(1+x)])/(2*x*(2-7*x)) + O[x]^30, x] // Rest (* Jean-François Alcover, Jun 19 2015 *)

{a(n) = local(A); if( n<1, 0, A = O(x); for( k=1, ceil(n/2), A = 1/( 1/x - 2 - (2 - 7*x) / (1 - 3*x) * A)); polcoeff(A, n))} /* Michael Somos, Apr 17 2012 */

G.f.: ((1-2x)(1-3x)-(1-4x)sqrt((1-3x)(1+x)))/(2x(2-7x)).
G.f. A(x) satisfies 0 = f(x, A(x)) where f(x ,y) = (7*x^2 - 2*x) * y^2 + (6*x^2 - 5*x + 1) * y + (3*x^2 - x). - Michael Somos, Apr 17 2012
0 = (105*n^2 + 861*n) * a(n) + (40*n^2 + 433*n + 672) * a(n+1) - (55*n^2 + 586*n + 1200) * a(n+2) + (10*n^2 + 112*n + 288) * a(n+3). - Michael Somos, Apr 17 2012
BINOMIAL transform is A059714. HANKEL transform is A001519(n+1). - Michael Somos, Apr 17 2012

More terms from James Sellers, Feb 09 2001

cp's OEIS Frontend

A026737 a(n) = T(2*n,n), T given by A026736.

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