A122391 -id:A122391 - cp's OEIS frontend

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).

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

Mike Zabrocki, Aug 31 2006

			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.

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. A118264, A122367, A122391, A122393, A122394.

coeffs(convert(series(mul(1-q^i,i=1..3)/(1-3*q),q,20),`+`)-O(q^20),q);

G.f.: (1-q)*(1-q^2)*(1-q^3)/(1-3*q) 3^n - 3^(n-1) - 3^(n-2) + 3^(n-4) + 3^(n-5) - 3^(n-6) (for n>5) a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 15, a(4) = 46, a(5) = 139, a(n) = 416*3^(n-6) for n>5

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

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).

Original entry on oeis.org

1, 4, 19, 95, 475, 2376, 11881, 59406, 297029, 1485144, 7425719, 37128595, 185642975, 928214876, 4641074381, 23205371904, 116026859520, 580134297600, 2900671488000, 14503357440000, 72516787200000, 362583936000000

Offset: 0

Mike Zabrocki, Aug 31 2006

			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

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. A118266, A122369, A122391, A122392, A122393.

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

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

A176414 Expansion of (7+8x)/(1+2x).

Original entry on oeis.org

7, -6, 12, -24, 48, -96, 192, -384, 768, -1536, 3072, -6144, 12288, -24576, 49152, -98304, 196608, -393216, 786432, -1572864, 3145728, -6291456, 12582912, -25165824, 50331648, -100663296, 201326592, -402653184, 805306368

Offset: 0

Klaus Brockhaus, Apr 17 2010

Inverse binomial transform of A176415.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2).

Cf. A176415, A110164 (essentially the same), A122803.

Join[{7},NestList[-2#&,-6,40]] (* Harvey P. Dale, Jun 20 2020 *)

{for(n=0, 29, print1(polcoeff((7+8*x)/(1+2*x)+x*O(x^n), n), ", "))}

A176414(n)=3*(-2)^n+!n*4 \\ M. F. Hasler, Apr 19 2015

a(n) = A110164(n+2) for n > 0.
a(n) = 3*(-2)^n = 3*A122803(n+1) for n > 0; a(0) = 7.
a(n) = -2*a(n-1) for n > 1; a(0) = 7, a(1) = -6.
a(n) = (-1)^n*A132477(n) = (-1)^n*A122391(n+3), n>1.
a(n) = (-1)^n*A111286(n+2) = (-1)^n*A098011(n+4) = (-1)^n*A091629(n) = (-1)^n*A087009(n+3) = (-1)^n*A082505(n+1) = (-1)^n*A042950(n+1) = (-1)^n*A007283(n) = (-1)^n*A003945(n+1), n>0. - R. J. Mathar, Dec 10 2010
E.g.f.: 4 + 3*exp(-2*x). - Alejandro J. Becerra Jr., Feb 15 2021

Edited by M. F. Hasler, Apr 19 2015

A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 4, 14, 16, 7, 1, 5, 25, 41, 29, 9, 1, 6, 41, 91, 92, 46, 11, 1, 7, 63, 182, 246, 175, 67, 13, 1, 8, 92, 336, 582, 550, 298, 92, 15, 1, 9, 129, 582, 1254, 1507, 1079, 469, 121, 17, 1, 10, 175, 957, 2508, 3718, 3367, 1925, 696, 154

Offset: 0

Philippe Deléham, Jan 24 2014

Triangle T(n,k), read by rows, given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are A111282(n+1) = A025169(n-1).
Diagonal sums are A122391(n+1) = A003945(n-1).

			Triangle begins:
  1;
  1,  1;
  2,  3,   1;
  3,  7,   5,   1;
  4, 14,  16,   7,   1;
  5, 25,  41,  29,   9,  1;
  6, 41,  91,  92,  46, 11,  1;
  7, 63, 182, 246, 175, 67, 13, 1;

Cf. Columns: A028310, A004006.
Cf. Diagonals: A000012, A005408, A130883.
Cf. Similar sequences: A078812, A085478, A111125, A128908, A165253, A207606.
Cf. A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare(1+x/(1-x)^2, 8); # Peter Luschny, Mar 06 2022

CoefficientList[#, y] & /@
CoefficientList[
Series[(1 - x + x^2)/(1 - 2*x - x*y + x^2), {x, 0, 12}], x] (* Wouter Meeussen, Jan 25 2014 *)

G.f.: (1 - x + x^2)/(1 - 2*x - x*y + x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(2,2) = 1, T(n,k) = 0 if k < 0 or k > n.
The Riordan square (see A321620) of 1 + x/(1 - x)^2. - Peter Luschny, Mar 06 2022

A369584 a(n) = 2^(n - 3) - A368279(n) for n >= 5, otherwise 0. Number of compositions of n whose first and last part is not equal to 1 and whose first part is not the largest part.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 6, 13, 30, 65, 140, 296, 622, 1294, 2679, 5518, 11323, 23160, 47250, 96184, 195438, 396490, 803292, 1625591, 3286340, 6637913, 13397224, 27020974, 54465702, 109725932, 220944768, 444700208, 894701728, 1799419458, 3617792587, 7271505058

Offset: 0

Peter Luschny, Jan 29 2024

nonn

We consider a tripartition of the compositions of an integer n >= 1, which we sort in lexicographic order. For this purpose, we define three predicates for a composition C.
  (1) The first part of C is the largest part of C.
  (2) The first part of C is 1.
  (3) The last part of C is 1.
We thus define three classes of compositions:
  X(n): compositions for which (1) and not (3) applies;
  Y(n): compositions for which (2) or (3) applies;
  Z(n): compositions for which not (1) and not (2) and not (3) applies.
X(n), Y(n), and Z(n) are disjoint classes whose union comprises all compositions of n; they thus form a disjoint tripartition of the compositions of n. The number of these compositions are given by:
  card(Z(n)) = a(n); card(X(n)) = A368279(n); card(Y(n)) = A122391(n).

			The compositions of class Z(n) for n = 5..7 are:
  5: [2, 3];
  6: [2, 1, 3], [2, 4];
  7: [2, 1, 1, 3], [2, 1, 4], [2, 2, 3], [2, 3, 2], [2, 5], [3, 4].
The cardinalities of some tripartitions, |X(n)| + |Y(n)| + |Z(n)| = 2^(n - 1):
   5:  12 +  3 +  1 =  16;
   6:  24 +  6 +  2 =  32;
   7:  48 + 10 +  6 =  64;
   8:  96 + 19 + 13 = 128;
   9: 192 + 34 + 30 = 256;
  10: 384 + 63 + 65 = 512.

Cf. A011782, A122391, A368279.

gf := 1 + (x - 1)*(1 + x^2 / (2*x - 1) + sum(x^n / (1 - x*(1 - x^n)/(1 - x)),
n = 1..42)): ser := series(gf, x, 40): seq(coeff(ser, x, n), n = 0..36);

def A369584(n):
    if n < 5: return 0
    return 2^(n - 3) - A368279(n)
print([A369584(n) for n in range(37)])

a(n) = [x^n] (1 + (x - 1)*(1 + x^2/(2*x - 1) + Sum_{k>=1} x^k/(1 - x*(1 - x^k)/(1 - x)))).

card(X(n)) + card(Y(n)) + card(Z(n)) = A011782(n) = 2^(n - 1) for n > 0.

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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).

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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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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).

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A176414 Expansion of (7+8*x)/(1+2*x).

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A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

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A369584 a(n) = 2^(n - 3) - A368279(n) for n >= 5, otherwise 0. Number of compositions of n whose first and last part is not equal to 1 and whose first part is not the largest part.

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A176414 Expansion of (7+8x)/(1+2x).