A091665 -id:A091665 - cp's OEIS frontend

A046649 a(n) is the number of nonseparable planar maps with 2*n+1 edges and a fixed outer face of 4 edges which are invariant under a rotation of a 1/2 turn. (Column 2 of A091665.)

2, 8, 34, 160, 806, 4256, 23256, 130416, 746350, 4341480, 25594530, 152585472, 918324904, 5572034240, 34048494608, 209347674768, 1294227005694, 8040125464280, 50165404177350, 314229490307040, 1975283452131990, 12456968750889600, 78790615438385760, 499700263517332800

Offset: 2

N. J. A. Sloane

Andrew Howroyd, Table of n, a(n) for n = 2..200
W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

Column 2 of A091665.

a(n) = 4*(7*n-11)*(3*n-5)!/((n-2)!*(2*n-1)!). - Emeric Deutsch, Mar 03 2004

G.f.: 2*(g+1)/(1-g)^3 where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

More terms from Emeric Deutsch, Mar 03 2004

Terms a(23) and beyond from Andrew Howroyd, Mar 29 2021

A046652 Triangle of rooted planar maps, read by rows.

Original entry on oeis.org

1, 2, 2, 3, 8, 7, 4, 21, 34, 30, 5, 44, 114, 160, 143, 6, 80, 308, 609, 806, 728, 7, 132, 715, 1908, 3315, 4256, 3876, 8, 203, 1482, 5185, 11420, 18444, 23256, 21318, 9, 296, 2814, 12600, 34520, 67856, 104652, 130416, 120175, 10, 414, 4984, 27965, 93924, 221300, 404016, 603801, 746350, 690690, 11, 560, 8343, 57584, 234066, 654336, 1394505, 2418372, 3533145, 4341480, 4032015

Offset: 0

N. J. A. Sloane

			Triangle begins:
  1;
  2,   2;
  3,   8,   7;
  4,  21,  34,  30;
  5,  44, 114, 160, 143;
  6,  80, 308, 609, 806, 728;
  ...

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]

A091665 is the same triangle with rows reversed and has much more information.

T := proc(n, k) if k<=n then k*sum((2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!, j=k..min(n, 2*k-1))/(2*n-k+1)! else 0 fi end: seq(seq(T(n, n-k+1), k=1..n), n=1..11); # Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

t[n_, k_] := If[k <= n, k*Sum[(2*j-k+1)*(j-1)!*(3*n-k-j)!/(j-k+1)!/(j-k)!/(2*k-j-1)!/(n-j)!, {j, k, Min[n, 2*k-1]}]/(2*n-k+1)!, 0]; Table[t[n, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 20 2014, after Herman Jamke *)

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

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

A046649 a(n) is the number of nonseparable planar maps with 2*n+1 edges and a fixed outer face of 4 edges which are invariant under a rotation of a 1/2 turn. (Column 2 of A091665.)

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A046652 Triangle of rooted planar maps, read by rows.

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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

A046649 a(n) is the number of nonseparable planar maps with 2*n+1 edges and a fixed outer face of 4 edges which are invariant under a rotation of a 1/2 turn. (Column 2 of A091665.)

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A046652 Triangle of rooted planar maps, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

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).