Markus Kuba - cp's OEIS frontend

User: Markus Kuba

Markus Kuba has authored 2 sequences.

A378026 Number of simple lattice paths, steps (-1,0,0),(0,-1,0),(0,0,-1), of length 3n from (n,n,n) to the origin, never returning to the diagonal x = y = z before the origin.

1, 6, 54, 816, 14814, 295812, 6262488, 137929392, 3125822238, 72383434332, 1704669773652, 40693683620448, 982302086191752, 23933136140685648, 587728374471479952, 14530886841268923264, 361374588105759096606, 9033515437023805672044, 226844689948433272890396, 5719461854507320708714464

Offset: 0

Markus Kuba, Nov 14 2024

Inversion of A006480 de Bruijn's S(3,n): (3n)!/(n!)^3.

Alois P. Heinz, Table of n, a(n) for n = 0..701
Markus Kuba and Alois Panholzer, Lattice paths and the diagonal of the cube, arXiv:2411.03930 [math.CO], 2024.

Cf. A006480, A054474, A000984.

b:= proc(n) option remember; (3*n)!/(n!)^3 end:
a:= proc(n) option remember;
      b(n)-add(a(n-i)*b(i), i=1..n-1)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Nov 15 2024

nmax = 20; CoefficientList[Series[2 - 1/Hypergeometric2F1[1/3, 2/3, 1, 27*x], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 15 2024 *)

G.f.: 2 - 1/P(z) where P(z) = 2F1(1/3,2/3;1;27z).

INVERTi transform of A006480.

More terms from Vaclav Kotesovec, Nov 15 2024

A233389 Naturally embedded ternary trees having no internal node of label greater than 1.

Original entry on oeis.org

1, 1, 3, 11, 46, 209, 1006, 5053, 26227, 139726, 760398, 4211959, 23681987, 134869448, 776657383, 4516117107, 26486641078, 156532100029, 931426814462, 5576590927886, 33574649282538, 203169756237944, 1235156720288767, 7541099028832261, 46222213821431646

Offset: 0

Markus Kuba, Dec 08 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Markus Kuba, A note on naturally embedded ternary trees, Electronic Journal of Combinatorics, Volume 18 (1), paper P142, 2011.
Markus Kuba, A note on naturally embedded ternary trees, arXiv:0902.2646 [math.CO], 2009.

Cf. A000139, A001764.

a:= proc(n) option remember; `if`(n<3, 1+n*(n-1),
      ((1349*n^2-2738*n+953)*n*a(n-1) -(5567*n^3-20114*n^2
       +22439*n-7320)*a(n-2)-(3*(3*n-4))*(19*n-11)*(3*n-5)
       *a(n-3))/((2*(2*n-1))*(n+1)*(19*n-30)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 03 2017

a[n_] := a[n] = If[n < 3, 1 + n*(n - 1), ((1349*n^2 - 2738*n + 953)*n*a[n - 1] - (5567*n^3 - 20114*n^2 + 22439*n - 7320)*a[n - 2] - (3*(3*n - 4)) * (19*n - 11)*(3*n - 5)*a[n - 3])/((2*(2*n - 1))*(n + 1)*(19*n - 30))];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 09 2017, after Alois P. Heinz *)

N=66; x='x+O('x^N); T=serreverse(x-x^3)/x; v=Vec(((T-2)*T^3/(T^2-3*T+1))); vector(#v\2, n, v[2*n-1]) \\ Joerg Arndt, May 26 2016

G.f.: (T(z) - 2)*T^3(z)/(T^2(z) - 3*T(z) + 1), where T(z) = 1 + z*T^3(z) is the generating function of ternary trees - see A001764.
From Peter Bala, Feb 06 2022: (Start)
a(n) = (2/(n+1))*binomial(3*n,n) + Sum_{k=0..n} (-1)^(k+1)*Fibonacci(k+1)* binomial(3*n,n-k)*(n*(11*k+5)-2*k(k+1))/(n*(2*n+k+1)) for n >= 1. See Kuba, Corollary 1, p. 6.
O.g.f.: A(x) = (1/x)*(B(x) - 2)/(B(x) - 1), where B(x) = Sum_{n >= 0} 2*(3*n)!/((2*n+1)!*((n+1)!))*x^n is the o.g.f. of A000139. (End)

More terms from F. Chapoton, May 26 2016

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User: Markus Kuba

A378026 Number of simple lattice paths, steps (-1,0,0),(0,-1,0),(0,0,-1), of length 3n from (n,n,n) to the origin, never returning to the diagonal x = y = z before the origin.

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