A320064 The number of F_2 graphs on { 1, 2, ..., n } with a unique cycle of weight 1, which corresponds to the number of reflectable bases of the root system of type D_n.

0, 1, 16, 312, 7552, 220800, 7597824, 301321216, 13545271296, 681015214080, 37879390720000, 2309968030334976, 153275504883695616, 10995166075754119168, 847974316241667686400, 69971459959477921382400, 6151490510604350965940224, 574035430519008722436489216, 56669921387839814123670994944

Offset: 1

Author

Masaya Tomie, Oct 04 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..350
• Federico Ardila, Matthias Beck, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
• S. Azam, M. B. Soltani, M. Tomie and Y. Yoshii, A graph theoretical classification for reflectable bases, PRIMS, Vol 55 no 4, (2019), 689-736.
• Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, Hurwitz numbers for reflection groups III: Uniform formulas, arXiv:2308.04751 [math.CO], 2023, see p. 11.

Crossrefs

Cf. A000435, A001863, A038057.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[(&+[j^(j-1)*(2*x)^j/Factorial(j): j in [1..m+2]])^k/(4*k): k in [2..m+1]]) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Dec 10 2018

Mathematica

nmax = 20; Rest[CoefficientList[Series[Sum[1/(4*m)*(Sum[k^(k-1)*(2*x)^k/k!, {k, 1, nmax}])^m, {m, 2, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Oct 23 2018 *)

PARI

seq(n)={Vec(serlaplace(sum(m=2, n, (sum(k=1, n, k^(k-1)*(2*x)^k/k!) + O(x^n))^m/(4*m))), -n)} \\ Andrew Howroyd, Nov 07 2018

PARI

apply( A320064(n)=A001863(n)*(n-1)<<(n-2), [1..20]) \\ Defines the function A320064. The additional apply(...) provides a check and illustration. - M. F. Hasler, Dec 09 2018

Python

from math import comb
def A320064(n): return 0 if n<2 else ((sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n))//n<Chai Wah Wu, Apr 25-26 2023

Formula

E.g.f.: Sum_{m>=2} (1/(4*m)) (Sum_{k>=1} k^(k-1)*(2*x)^k/k!)^m.

a(n) = (n-1)*2^(n-2)*A001863(n). - M. F. Hasler, Dec 09 2018

a(n) = 2^(n-2)*A000435(n). - Chai Wah Wu, Apr 25 2023