A118265
Coefficient of q^n in (1-q)^4/(1-4q); dimensions of the enveloping algebra of the derived free Lie algebra on 4 letters.
Original entry on oeis.org
1, 0, 6, 20, 81, 324, 1296, 5184, 20736, 82944, 331776, 1327104, 5308416, 21233664, 84934656, 339738624, 1358954496, 5435817984, 21743271936, 86973087744, 347892350976, 1391569403904, 5566277615616, 22265110462464, 89060441849856
Offset: 0
The enveloping algebra of the derived free Lie algebra is characterized as the intersection of the kernels of all partial derivative operators in the space of non-commutative polynomials, a(0) = 1 since all constants are killed by derivatives, a(1) = 0 since no polys of degree 1 are killed, a(2) = 6 since all Lie brackets [x1,x2], [x1,x3], [x1, x4], [x2,x3], [x2,x4], [x3,x4] are killed by all derivative operators.
- 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.
- Michael De Vlieger, Table of n, a(n) for n = 0..1661
- Nantel Bergeron, Christophe Reutenauer, Mercedes Rosas, and Mike Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082, 2005. See also Canad. J. Math. 60 (2008), no. 2, 266-296.
- Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See p. 21.
- Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
- Index entries for linear recurrences with constant coefficients, signature (4).
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f:=n->add((-1)^k*C(4,k)*4^(n-k),k=0..min(n,4)); seq(f(i),i=0..15);
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a[n_] := If[n<4, {1, 0, 6, 20}[[n+1]], 81*4^(n-4)];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Dec 10 2018 *)
A118266
Coefficient of q^n in (1-q)^5/(1-5q); dimensions of the enveloping algebra of the derived free Lie algebra on 5 letters.
Original entry on oeis.org
1, 0, 10, 40, 205, 1024, 5120, 25600, 128000, 640000, 3200000, 16000000, 80000000, 400000000, 2000000000, 10000000000, 50000000000, 250000000000, 1250000000000, 6250000000000, 31250000000000, 156250000000000
Offset: 0
- 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.
- Michael De Vlieger, Table of n, a(n) for n = 0..1431
- Nantel Bergeron, Christophe Reutenauer, Mercedes Rosas, and Mike Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082, 2005. See also Canad. J. Math. 60 (2008), no. 2, 266-296.
- Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See p. 21.
- Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
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f:=n->add((-1)^k*binomial(5,k)*5^(n-k),k=0..min(n,4)): seq(f(i),i=0..15);
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a[n_] := If[n<6, {1, 0, 10, 40, 205, 1024}[[n+1]], 1024*5^(n-5)];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 10 2018 *)
A122392
Dimension of 3-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 3 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).
Original entry on oeis.org
1, 2, 5, 15, 46, 139, 416, 1248, 3744, 11232, 33696, 101088, 303264, 909792, 2729376, 8188128, 24564384, 73693152, 221079456, 663238368, 1989715104, 5969145312, 17907435936, 53722307808, 161166923424, 483500770272, 1450502310816
Offset: 0
a(1) = 2 because x1 - x2, x2 - x3 are killed by d_x1 + d_x2 + d_x3
a(2) = 5 because x1 x2 - x2 x1, x1 x3 - x3 x1, x2 x3 - x3 x2, 2 x1 x2 - x2 x2 - 2 x1 x3 + x3 x3,
x1 x1 - 2 x2 x1 + 2 x2 x3 - x3 x3 are killed by d_x1 + d_x2 + d_x3, d_x1^2 + d_x2^2 + d_x3^2 and
d_x1 d_x2 + d_x1 d_x3 + d_x2 d_x3
- 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.
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coeffs(convert(series(mul(1-q^i,i=1..3)/(1-3*q),q,20),`+`)-O(q^20),q);
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