A321233 - OEIS

%I #58 Apr 26 2023 18:34:01

%S 0,4,128,4992,241664,14131200,972521472,77138231296,6935178903552,

%T 697359579217920,77576992194560000,9461629052252061696,

%U 1255632936007234486272,180144800985155488448512,27786422394606966747955200,4585649599904345055716966400,806288164205933489807717040128

%N a(n) is the number of reflectable bases of the root system of type D_n.

%C The root systems of type D_n are only defined for n >= 4. See chapter 3 of the Humphreys reference. Sequence extended to n=1 using formula/recurrence.

%D J. E. Humphreys, Introduction to Lie algebras and representation theory, 2nd ed, Springer-Verlag, New York, 1972.

%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.

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

%F a(n) = 2^n*A320064(n).

%F a(n) = (n-1)*4^(n-1)*A001863(n). - _M. F. Hasler_, Dec 09 2018

%t Rest[With[{m = 25}, CoefficientList[Series[Sum[Sum[j^(j - 1)*(4*x)^j/j!, {j, 1, m + 1}]^k/(4*k), {k, 2, m}], {x, 0, m}], x]*Range[0, m]!]] (* _G. C. Greubel_, Dec 09 2018 *)

%o (PARI) a(n)={n!*polcoef(sum(m=2, n, (sum(k=1, n, k^(k-1)*(4*x)^k/k!) + O(x^(n-m+2)))^m/(4*m)), n)} \\ _Andrew Howroyd_, Nov 01 2018

%o (PARI) A321233(n)=A001863(n)*(n-1)*4^(n-1) \\ _M. F. Hasler_, Dec 09 2018

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[ (&+[ j^(j-1)*(4*x)^j/Factorial(j) :j in [1..m+3]])^k/(4*k) :k in [2..m+2]]) )); [0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // _G. C. Greubel_, Dec 09 2018

%o (Python)

%o from math import comb

%o def A321233(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-1<<1) # _Chai Wah Wu_, Apr 26 2023

%Y Cf. A000435, A320064.

%K nonn

%O 1,2

%A _Masaya Tomie_, Nov 01 2018