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%I #22 Sep 01 2023 02:22:55
%S 1,2,8,80,2080,158080,36674560,28019363840,73410733260800,
%T 697108323044556800,24883978699398499532800,
%U 3487539382678098506520985600,1982680089210029713351206397542400,4739557099654791829171791869197156352000
%N Number of matchings in the n-triangular honeycomb rook graph.
%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="/A289897/b289897.txt">Table of n, a(n) for n = 1..50</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>
%F a(n) = Product_{k=1..n} A000085(k). - _Andrew Howroyd_, Jul 17 2017
%F log(a(n)) ~ n^2*log(n)/4 - 3*n^2/8 + 2*n^(3/2)/3. - _Vaclav Kotesovec_, Aug 29 2023
%t FoldList[Times, Table[HypergeometricPFQ[{-k/2, (1 - k)/2}, {}, 2], {k, 20}]] (* _Eric W. Weisstein_, Jul 19 2017 *)
%t Table[(-1/2)^(Binomial[n + 1, 2]/2) Product[HermiteH[k, -I/Sqrt[2]], {k, n}], {n, 20}] (* _Eric W. Weisstein_, Jul 19 2017 *)
%t Table[Product[HypergeometricPFQ[{-k/2, (1 - k)/2}, {}, 2], {k, n}], {n, 20}] (* _Eric W. Weisstein_, Jul 19 2017 *)
%o (PARI)
%o a(n) = prod(k=1, n, k! * polcoeff( exp( x + x^2 / 2 + x * O(x^k)), k)); \\ _Andrew Howroyd_, Jul 17 2017
%o (Python)
%o from math import prod, factorial
%o def A289897(n): return prod(sum(factorial(k)//(factorial(k-(m<<1))*factorial(m)*(1<<m)) for m in range((k>>1)+1)) for k in range(1,n+1)) # _Chai Wah Wu_, Aug 31 2023
%Y Cf. A289900.
%K nonn
%O 1,2
%A _Eric W. Weisstein_, Jul 14 2017
%E Terms a(11) and beyond from _Andrew Howroyd_, Jul 17 2017
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