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
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