OFFSET
1,2
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.
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
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 *)
PROG
(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
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Masaya Tomie, Nov 01 2018
STATUS
approved