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%I #41 Aug 29 2023 10:26:41
%S 1,1,3,9,135,2025,212625,22325625,21097715625,19937341265625,
%T 207248662456171875,2154349846231906640625,
%U 291128066470548703880859375,39341591262497599098939931640625,79746389028864195813528714933837890625
%N Number of maximal matchings in the n-triangular honeycomb rook graph.
%C Also the number of maximum matchings for n > 1.
%C 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
%H Andrew Howroyd, <a href="/A289900/b289900.txt">Table of n, a(n) for n = 1..50</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximalIndependentEdgeSet.html">Maximal Independent Edge Set</a>
%F a(n) = Product_{k=1..n} A001147(ceiling(k/2)). - _Andrew Howroyd_, Jul 17 2017
%F 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
%t MapAt[# - 1 &, #, 1] &@ FoldList[Times, Array[(2 Ceiling[#/2] - 1)!! &, 15]] (* _Michael De Vlieger_, Jul 18 2017 *)
%t FoldList[Times, Table[(k - Mod[k - 1, 2])!!, {k, 15}]] (* _Eric W. Weisstein_, Jul 19 2017 *)
%t Table[Product[(k - Mod[k - 1, 2])!!, {k, n}], {n, 15}] (* _Eric W. Weisstein_, Jul 19 2017 *)
%t 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 *)
%o (PARI)
%o a(n)=prod(k=1,n, k!/((k\2)!*2^(k\2))); \\ _Andrew Howroyd_, Jul 17 2017
%o (Python)
%o from sympy import factorial2, ceiling
%o from operator import mul
%o def a001147(n):
%o return factorial2(2*n - 1)
%o def a(n):
%o return reduce(mul, [a001147(ceiling(k/2)) for k in range(1, n + 1)])
%o print([a(n) for n in range(1, 31)]) # _Indranil Ghosh_, Jul 18 2017, after PARI code
%Y Cf. A289897.
%K nonn
%O 1,3
%A _Eric W. Weisstein_, Jul 14 2017
%E Terms a(11) and beyond from _Andrew Howroyd_, Jul 17 2017
%E a(1) changed to 1 by _N. J. A. Sloane_, Jul 18 2017
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