A289900 Number of maximal matchings in the n-triangular honeycomb rook graph.

1, 1, 3, 9, 135, 2025, 212625, 22325625, 21097715625, 19937341265625, 207248662456171875, 2154349846231906640625, 291128066470548703880859375, 39341591262497599098939931640625, 79746389028864195813528714933837890625

Offset

1,3

Comments

Also the number of maximum matchings for n > 1.

The n-triangular honeycomb rook graph is the disjoint union of the complete graphs K_k for k in {1..n}. In terms of a triangular chessboard it is the graph for a chesspiece that is constrained to move on a single axis. - Andrew Howroyd, Jul 17 2017

Links

Andrew Howroyd, Table of n, a(n) for n = 1..50

Eric Weisstein's World of Mathematics, Matching

Eric Weisstein's World of Mathematics, Maximal Independent Edge Set

Formula

a(n) = Product_{k=1..n} A001147(ceiling(k/2)). - Andrew Howroyd, Jul 17 2017

a(n) ~ A * 2^(1/3 + n/2) * n^(1/(15/2 + 9*(-1)^n/2) + n/2 + n^2/4) / exp(1/12 + n/2 + 3*n^2/8), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023

Mathematica

MapAt[# - 1 &, #, 1] &@ FoldList[Times, Array[(2 Ceiling[#/2] - 1)!! &, 15]] (* Michael De Vlieger, Jul 18 2017 *)
FoldList[Times, Table[(k - Mod[k - 1, 2])!!, {k, 15}]] (* Eric W. Weisstein, Jul 19 2017 *)
Table[Product[(k - Mod[k - 1, 2])!!, {k, n}], {n, 15}] (* Eric W. Weisstein, Jul 19 2017 *)
Table[2^(n (n + 2)/4 - 1/12)  E^(-1/4) Pi^(-(n + 1)/2) Glaisher^3 If[Mod[n, 2] == 0, BarnesG[(3 + n)/2]^2, 2^(1/4) BarnesG[n/2 + 1] BarnesG[n/2 + 2]], {n, 15}] (* Eric W. Weisstein, Jul 19 2017 *)

Prog

(PARI)
a(n)=prod(k=1, n, k!/((k\2)!*2^(k\2))); \\ Andrew Howroyd, Jul 17 2017
(Python)
from sympy import factorial2, ceiling
from operator import mul
def a001147(n):
    return factorial2(2*n - 1)
def a(n):
    return reduce(mul, [a001147(ceiling(k/2)) for k in range(1, n + 1)])
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Jul 18 2017, after PARI code

Crossrefs

Cf. A289897.

Sequence in context: A368092 A163402 A288757 * A235062 A082707 A087193

Adjacent sequences: A289897 A289898 A289899 * A289901 A289902 A289903

Keyword

nonn

Author

Eric W. Weisstein, Jul 14 2017

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jul 17 2017

a(1) changed to 1 by N. J. A. Sloane, Jul 18 2017

Status

approved