A321233 a(n) is the number of reflectable bases of the root system of type D_n.

0, 4, 128, 4992, 241664, 14131200, 972521472, 77138231296, 6935178903552, 697359579217920, 77576992194560000, 9461629052252061696, 1255632936007234486272, 180144800985155488448512, 27786422394606966747955200, 4585649599904345055716966400, 806288164205933489807717040128

Offset: 1

Author

Masaya Tomie, Nov 01 2018

Keywords

• nonn

Comments

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.

References

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

Links

• 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.

Crossrefs

Cf. A000435, A320064.

Programs

Magma

m:=25; R:=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

Mathematica

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 *)

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

PARI

A321233(n)=A001863(n)*(n-1)*4^(n-1) \\ M. F. Hasler, Dec 09 2018

Python

from math import comb
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

Formula

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

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

a(n) = (n-1)*4^(n-1)*A001863(n). - M. F. Hasler, Dec 09 2018