A118265 -id:A118265 - cp's OEIS frontend

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.

1, 0, 10, 40, 205, 1024, 5120, 25600, 128000, 640000, 3200000, 16000000, 80000000, 400000000, 2000000000, 10000000000, 50000000000, 250000000000, 1250000000000, 6250000000000, 31250000000000, 156250000000000

Offset: 0

Mike Zabrocki, Apr 20 2006

nonn

For n>=5, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5} such that for fixed, different x_1, x_2, x_3, x_4, x_5 in {1,2,...,n} and fixed y_1, y_2, y_3, y_ 4, y_5 in {1,2,3,4,5} we have f(x_i)<>y_i, (i=1,2,3,4,5). - Milan Janjic, May 13 2007

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

Cf. A001692, A118264, A118265.

f:=n->add((-1)^k*binomial(5,k)*5^(n-k),k=0..min(n,4)): seq(f(i),i=0..15);

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 *)

G.f.: (1-q)^5/(1-5q) sum( (-1)^k*C(5,k) 5^(n-k); k=0..min(n,5));

a(n) = 1024*5^(n-5) for n>5. - Jean-François Alcover, Dec 10 2018

A118264 Coefficient of q^n in (1-q)^3/(1-3q); dimensions of the enveloping algebra of the derived free Lie algebra on 3 letters.

Original entry on oeis.org

1, 0, 3, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 472392, 1417176, 4251528, 12754584, 38263752, 114791256, 344373768, 1033121304, 3099363912, 9298091736, 27894275208, 83682825624, 251048476872, 753145430616

Offset: 0

Mike Zabrocki, Apr 20 2006

a(n) is the number of generalized compositions of n when there are i^2-1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

			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) = 3 since all Lie brackets [x1,x2], [x1,x3], [x2, x3] 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..2097
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 pp. 17, 20.
Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A080923, A027376, A118265, A118266.

f:=n->coeftayl((1-q)^3/(1-3*q),q=0,n):seq(f(i),i=0..15);

CoefficientList[Series[(1-q)^3/(1-3q),{q,0,30}],q] (* or *) Join[{1,0,3}, NestList[3#&,8,30]] (* Harvey P. Dale, Jun 28 2011 *)
Join[{1, 0, 3}, LinearRecurrence[{3}, {8}, 24]] (* Jean-François Alcover, Sep 23 2017 *)

G.f.: (1-x)^3/(1-3x).
a(n) = 3^{n-1}-3^{n-3} for n>=3.
a(n) = A080923(n-1), n>1.
If p[i]=i^2-1 and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - Milan Janjic, May 02 2010
For a(n)>=8, a(n+1)=3*a(n). - Harvey P. Dale, Jun 28 2011

Formula corrected Mike Zabrocki, Jul 22 2010

A122393 Dimension of 4-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 4 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, 3, 11, 44, 176, 706, 2824, 11296, 45183, 180731, 722925, 2891700, 11566800, 46267200, 185068800, 740275200, 2961100800, 11844403200, 47377612800, 189510451200, 758041804800, 3032167219200, 12128668876800, 48514675507200

Offset: 0

Mike Zabrocki, Aug 31 2006

			a(1) = 3 because x1 - x2, x2 - x3, x3 - x4 are all killed by d_x1+d_x2+d_x3+d_x4

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.

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.

Cf. A118265, A122368, A122391, A122392, A122394.

coeffs(convert(series(mul(1-q^i,i=1..4)/(1-4*q),q,20),`+`)-O(q^20),q);

G.f.: (1-q)*(1-q^2)*(1-q^3)*(1-q^4)/(1-4*q) a(n) = 722925*4^(n-10) for n>9

A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

Original entry on oeis.org

1, 3, 4, 9, 12, 16, 27, 36, 48, 64, 81, 108, 144, 192, 256, 243, 324, 432, 576, 768, 1024, 729, 972, 1296, 1728, 2304, 3072, 4096, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 6561, 8748, 11664, 15552, 20736, 27648, 36864, 49152, 65536, 19683, 26244, 34992, 46656, 62208, 82944, 110592, 147456, 196608, 262144

Offset: 0

Wolfdieter Lang, Aug 01 2022

This is Boethius's triangle, with rows read as columns. See the link and reference.

			The triangle T begins:
n\k     0     1      2      3      4      5      6      7      8      9  ...
0:      1
1:      3     4
2:      9    12     16
3:     27    36     48     64
4:     81   108    144    192    256
5:    243   324    432    576    768   1024
6:    729   972   1296   1728   2304   3072   4096
7:   2187  2916   3888   5184   6912   9216  12288  16384
8:   6561  8748  11664  15552  20736  27648  36864  49152  65536
9:  19683 26244  34992  46656  62208  82944 110592 147456 196608 262144
...

Thomas Sonar, 3000 Jahre Analysis, 2. Auflage, Springer Spektrum, 2016, p.94, Abb. 3.1.2 und Abb. 3.1.3.

Anicius Manlius Severinus Boethius, De Institutione Arithmetica, 1488, p. 55 of 104, top of left column.

Columns: A000244, A003946, A257970, ...
Diagonals: A000302, A002001(n+1), A002063, A002063(n+3), A118265(n+4), ...
Row sums: A005061(n+1).

T[n_, k_] := 3^(n - k) * 4^k; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 05 2022 *)

T(n, k) = 3^(n-k)*4^k, for n >= 0, and k = 1, 2, ..., n.

G.f. of row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k: G(x, y) = 1/((1 - 3*x)*(1 - 4*x*y)).

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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.

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A118264 Coefficient of q^n in (1-q)^3/(1-3q); dimensions of the enveloping algebra of the derived free Lie algebra on 3 letters.

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A122393 Dimension of 4-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 4 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).

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A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

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