A289897 -id:A289897 - cp's OEIS frontend

A289900 Number of maximal matchings in the n-triangular honeycomb rook graph.

1, 1, 3, 9, 135, 2025, 212625, 22325625, 21097715625, 19937341265625, 207248662456171875, 2154349846231906640625, 291128066470548703880859375, 39341591262497599098939931640625, 79746389028864195813528714933837890625

Offset: 1

Eric W. Weisstein, Jul 14 2017

nonn

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

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

Cf. A289897.

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 *)

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

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

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

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

a(1) changed to 1 by N. J. A. Sloane, Jul 18 2017

A365269 a(n) = Product_{k=1..n} A002720(k).

Original entry on oeis.org

1, 2, 14, 476, 99484, 153802264, 2049722772328, 268353804798726416, 386893462638663037013264, 6798536031341327693983294520096, 1595359632648441879172205168815801694176, 5432770180592069558569584672506997142250856260032

Offset: 0

Vaclav Kotesovec, Aug 30 2023

nonn

Cf. A002720, A289897.

Table[Product[k! * LaguerreL[k,-1], {k,1,n}], {n,0,15}]
Table[BarnesG[n+2] * Product[LaguerreL[k,-1], {k,1,n}], {n,0,15}]

from math import prod, factorial, comb
def A365269(n): return prod(sum(factorial(m)*comb(k,m)**2 for m in range(k+1)) for k in range(1,n+1)) # Chai Wah Wu, Aug 31 2023

log(a(n)) ~ log(BarnesG(n+2)) + 4*n^(3/2)/3 - n*log(n)/4 - (1/4 + log(2) + log(Pi)/2)*n + 55*sqrt(n)/24.

log(a(n)) ~ n^2*log(n)/2 - 3*n^2/4 + 4*n^(3/2)/3 + 3*n*log(n)/4 - (5/4 + log(2)/2)*n + 55*sqrt(n)/24.

cp's OEIS Frontend

A289900 Number of maximal matchings in the n-triangular honeycomb rook graph.

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