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A289900 Number of maximal matchings in the n-triangular honeycomb rook graph. 2
1, 1, 3, 9, 135, 2025, 212625, 22325625, 21097715625, 19937341265625, 207248662456171875, 2154349846231906640625, 291128066470548703880859375, 39341591262497599098939931640625, 79746389028864195813528714933837890625 (list; graph; refs; listen; history; text; internal format)
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

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: A141143 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

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Last modified September 29 07:07 EDT 2022. Contains 357082 sequences. (Running on oeis4.)