A320064 - OEIS

%I #68 Aug 15 2023 17:01:55

%S 0,1,16,312,7552,220800,7597824,301321216,13545271296,681015214080,

%T 37879390720000,2309968030334976,153275504883695616,

%U 10995166075754119168,847974316241667686400,69971459959477921382400,6151490510604350965940224,574035430519008722436489216,56669921387839814123670994944

%N 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.

%H Vaclav Kotesovec, <a href="/A320064/b320064.txt">Table of n, a(n) for n = 1..350</a>

%H Federico Ardila, Matthias Beck, and Jodi McWhirter, <a href="https://arxiv.org/abs/2004.02952">The Arithmetic of Coxeter Permutahedra</a>, arXiv:2004.02952 [math.CO], 2020.

%H S. Azam, M. B. Soltani, M. Tomie and Y. Yoshii, <a href="https://doi.org/10.4171/PRIMS/55-4-2">A graph theoretical classification for reflectable bases</a>, PRIMS, Vol 55 no 4, (2019), 689-736.

%H Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, <a href="https://arxiv.org/abs/2308.04751">Hurwitz numbers for reflection groups III: Uniform formulas</a>, arXiv:2308.04751 [math.CO], 2023, see p. 11.

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

%F a(n) = (n-1)*2^(n-2)*A001863(n). - _M. F. Hasler_, Dec 09 2018

%F a(n) = 2^(n-2)*A000435(n). - _Chai Wah Wu_, Apr 25 2023

%t 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 *)

%o (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

%o (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

%o (Magma) m:=30; R<x>:=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

%o (Python)

%o from math import comb

%o 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<<n-2 # _Chai Wah Wu_, Apr 25-26 2023

%Y Cf. A000435, A001863, A038057.

%K nonn

%O 1,3

%A _Masaya Tomie_, Oct 04 2018