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A122394 Dimension of 5-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 5 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( w ) = sum over all subwords of w deleting xi once). 3
1, 4, 19, 95, 475, 2376, 11881, 59406, 297029, 1485144, 7425719, 37128595, 185642975, 928214876, 4641074381, 23205371904, 116026859520, 580134297600, 2900671488000, 14503357440000, 72516787200000, 362583936000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.

C. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. xviii+269 pp.

LINKS

Table of n, a(n) for n=0..21.

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082, Canad. J. Math. 60 (2008), no. 2, 266-296.

FORMULA

G.f.: (1-q)*(1-q^2)*(1-q^3)*(1-q^4)*(1-q^5)/(1-5*q) a(n) = 23205371904*5^(n-15) for n>14

EXAMPLE

a(1) = 4 because x1 - x2, x2 - x3, x3 - x4, x4 - x5 are all killed by d_x1+d_x2+d_x3+d_x4+d_x5

MAPLE

coeffs(convert(series(mul(1-q^i, i=1..5)/(1-5*q), q, 20), `+`)-O(q^20), q);

CROSSREFS

Cf. A118266, A122369, A122391, A122392, A122393.

Sequence in context: A027618 A278678 A020060 * A047781 A089354 A217217

Adjacent sequences:  A122391 A122392 A122393 * A122395 A122396 A122397

KEYWORD

nonn,easy

AUTHOR

Mike Zabrocki, Aug 31 2006

STATUS

approved

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Last modified February 18 00:19 EST 2019. Contains 320237 sequences. (Running on oeis4.)