A289897 Number of matchings in the n-triangular honeycomb rook graph.

1, 2, 8, 80, 2080, 158080, 36674560, 28019363840, 73410733260800, 697108323044556800, 24883978699398499532800, 3487539382678098506520985600, 1982680089210029713351206397542400, 4739557099654791829171791869197156352000

Offset: 1

Author

Eric W. Weisstein, Jul 14 2017

Keywords

• nonn

Comments

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, Independent Edge Set
• Eric Weisstein's World of Mathematics, Matching

Crossrefs

Cf. A289900.

Programs

Mathematica

FoldList[Times, Table[HypergeometricPFQ[{-k/2, (1 - k)/2}, {}, 2], {k, 20}]] (* Eric W. Weisstein, Jul 19 2017 *)
Table[(-1/2)^(Binomial[n + 1, 2]/2) Product[HermiteH[k, -I/Sqrt[2]], {k, n}], {n, 20}] (* Eric W. Weisstein, Jul 19 2017 *)
Table[Product[HypergeometricPFQ[{-k/2, (1 - k)/2}, {}, 2], {k, n}], {n, 20}] (* Eric W. Weisstein, Jul 19 2017 *)

PARI

a(n) = prod(k=1, n, k! * polcoeff( exp( x + x^2 / 2 + x * O(x^k)), k)); \\ Andrew Howroyd, Jul 17 2017

Python

from math import prod, factorial
def A289897(n): return prod(sum(factorial(k)//(factorial(k-(m<<1))*factorial(m)*(1<>1)+1)) for k in range(1,n+1)) # Chai Wah Wu, Aug 31 2023

Formula

a(n) = Product_{k=1..n} A000085(k). - Andrew Howroyd, Jul 17 2017

log(a(n)) ~ n^2*log(n)/4 - 3*n^2/8 + 2*n^(3/2)/3. - Vaclav Kotesovec, Aug 29 2023

Extensions

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