A046647 -id:A046647 - cp's OEIS frontend

A046646 a(n) = 2binomial(3n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.

1, 2, 6, 24, 110, 546, 2856, 15504, 86526, 493350, 2861430, 16829280, 100134216, 601661144, 3645533040, 22249511328, 136657509918, 844061090670, 5239262085330, 32665844580600, 204480219795390, 1284624902435490

Offset: 1

N. J. A. Sloane

Number of certain rooted planar maps.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 10.
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.
W. G. Brown, Enumeration of non-separable planar maps [Annotated scanned copy]

A diagonal of A046651.

Cf. A046647, A005470, A000139, A001764.

[1] cat [2*Binomial(3*n-3,n-1)/(2*n-1): n in [2..30]]; // Vincenzo Librandi, Oct 13 2013

alias(PS=ListTools:-PartialSums): A046646List := proc(m) local A, P, n;
A := [1,2]; P := [2]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A046646List(22); # Peter Luschny, Mar 26 2022

Join[{1},Table[(2*Binomial[3n-3,n-1])/(2n-1),{n,2,30}]] (* Harvey P. Dale, Oct 12 2013 *)

From Emeric Deutsch, Mar 03 2004: (Start)
a(n) = 2*binomial(3*n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.
a(n) = 2*A001764(n-1) for n >= 2. (End)
a(n) = (n+1) * A000139(n). - F. Chapoton, Feb 23 2024
G.f.: (1+g)/(1-g) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

More terms from Emeric Deutsch, Mar 03 2004

New name using a formula of Emeric Deutsch by Peter Luschny, Feb 23 2024

A091599 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with rn edges and a fixed outer face of rk edges which are invariant under a rotation of 1/r for any r >= 2 (independent of actual value of r).

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 24, 26, 12, 1, 110, 120, 75, 20, 1, 546, 594, 416, 174, 30, 1, 2856, 3094, 2289, 1176, 350, 42, 1, 15504, 16728, 12768, 7322, 2880, 636, 56, 1, 86526, 93024, 72420, 44388, 20475, 6324, 1071, 72, 1, 493350, 528770, 417240, 267240, 136252, 51495, 12740, 1700, 90, 1

Offset: 1

Emeric Deutsch, Mar 03 2004

Table I in the Brown reference.

			Triangle starts:
    1;
    2,   1;
    6,   6,  1;
   24,  26, 12,  1;
  110, 120, 75, 20, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

Column 1 gives A046646, column 2 gives A046647, row sums give A000259.
Same as A046651 but with rows reversed.
Cf. A046653, A091665.

T := proc(n,k) if k<=n then k*sum((2*j-k)*(j-1)!*(3*n-j-k-1)!/(j-k)!/(j-k)!/(2*k-j)!/(n-j)!,j=k..min(n,2*k))/(2*n-k)! else 0 fi end: seq(seq(T(n,k), k=1..n),n=1..11);

T(n, k) = k*sum(j=k, min(n, 2*k), (2*j-k)*(j-1)!*(3*n-j-k-1)!/(((j-k)!)^2*(2*k-j)!*(n-j)!))/(2*n-k)!
for(n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print) \\ Andrew Howroyd, Mar 29 2021

T(n, k) = k*(Sum_{j=k..min(n, 2*k)} (2*j-k)*(j-1)!*(3*n-j-k-1)!/(((j-k)!)^2*(2*k-j)!*(n-j)!))/(2*n-k)!

Name clarified by Andrew Howroyd, Mar 29 2021

cp's OEIS Frontend

A046646 a(n) = 2binomial(3n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.

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A091599 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with rn edges and a fixed outer face of rk edges which are invariant under a rotation of 1/r for any r >= 2 (independent of actual value of r).

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cp's OEIS Frontend

A046646 a(n) = 2*binomial(3*n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A091599 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with r*n edges and a fixed outer face of r*k edges which are invariant under a rotation of 1/r for any r >= 2 (independent of actual value of r).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A046646 a(n) = 2binomial(3n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.

A091599 Triangle read by rows: T(n,k) is the number of nonseparable planar maps with rn edges and a fixed outer face of rk edges which are invariant under a rotation of 1/r for any r >= 2 (independent of actual value of r).