A000259 Number of certain rooted planar maps. (Formerly M2943 N1185)

1, 3, 13, 63, 326, 1761, 9808, 55895, 324301, 1908878, 11369744, 68395917, 414927215, 2535523154, 15592255913, 96419104103, 599176447614, 3739845108057, 23435007764606, 147374772979438, 929790132901804, 5883377105975922, 37328490926964481, 237427707464042693

Offset

1,2

Comments

a(n) is also the number of North-East lattice paths from (0,0) to (2n,n) that begin with a North step, end with an East step, and do not bounce off the line y =1/2 x. Similarly, also the number of paths that begin with an East step, end with a North step, and do not bounce off the line y=1/2 x. - Michael D. Weiner, Aug 03 2017

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..1211

D. Birmajer, J. Gil, and M. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017.

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]

Formula

a(n) = Sum_{k = 1..n} (-1)^(k-1)*C(3n, n-k)*k/n*F(k-2) where F(k) is the k-th Fibonacci number (A000045) and F(-1) = 1. - Michael D. Weiner, Aug 03 2017

Example

For n = 2 the a(2) = 3 is counting the following three paths EEEENN, EEENEN, ENEEEN. The path EENEEN is excluded as it bounces off the line y = (1/2) x at the point (2, 1). - Michael D. Weiner, Aug 03 2017

Maple

with(linalg): T := proc(n, k) if k<=n then k*add((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: A := matrix(30, 30, T): seq(add(A[i, j], j=1..i), i=1..30); # Emeric Deutsch, Mar 03 2004
R := RootOf(x-t*(t-1)^2, t); ogf := series(1/((1-R-R^2)*(R-1)^2), x=0, 30); # Mark van Hoeij, Nov 08 2011

Mathematica

R = Root[#^3-2#^2+#-x&, 1]; CoefficientList[1/((1-R-R^2)*(R-1)^2) + O[x]^30, x] (* Jean-François Alcover, Feb 06 2016, after Mark van Hoeij *)
Table[Sum[(-1)^(k - 1)*Binomial[3 n, n - k]*k/n*Fibonacci[k - 2], {k, n}], {n, 21}] (* Michael De Vlieger, Aug 04 2017 *)

Prog

(PARI) a(n) = sum(k = 1, n, (-1)^(k-1)*binomial(3*n, n-k)*k/n*fibonacci(k-2)); \\ Michel Marcus, Aug 04 2017
(Magma) [&+[(-1)^(k-1)*Binomial(3*n, n-k)*k/n*Fibonacci(k - 2):k in [0..n]]: n in [1..30]]; // Vincenzo Librandi, Aug 05 2017
(Python)
from sympy import binomial, fibonacci
def a(n): return sum((-1)**(k - 1)*binomial(3*n, n - k)*k//n*fibonacci(k - 2) for k in range(1, n + 1))
print([a(n) for n in range(1, 21)]) # Indranil Ghosh, Aug 05 2017

Crossrefs

Row sums of A046651.

Sequence in context: A362744 A350519 A243280 * A007855 A193112 A367061

Adjacent sequences: A000256 A000257 A000258 * A000260 A000261 A000262

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Mar 03 2004

Status

approved