|
| |
|
|
A141419
|
|
A triangular sequence of coefficients of Dynkin diagram weights for the Cartan Groups B_n: t(n,m)=m*(2*n - m + 1)/2. As a rectangle, the accumulation array of A051340.
|
|
8
| |
|
|
1, 2, 3, 3, 5, 6, 4, 7, 9, 10, 5, 9, 12, 14, 15, 6, 11, 15, 18, 20, 21, 7, 13, 18, 22, 25, 27, 28, 8, 15, 21, 26, 30, 33, 35, 36, 9, 17, 24, 30, 35, 39, 42, 44, 45, 10, 19, 27, 34, 40, 45, 49, 52, 54, 55
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| Row sums are:
{1, 5, 14, 30, 55, 91, 140, 204, 285, 385};
Here all the weights are divided by two where they aren't in Cahn.
(Start) As a rectangle, A141419 is in the accumulation chain
...< A051340 < A141419 < A185874 < A185875 < A185876 < ...
(See A144112 for the definition of accumulation array.)
row 1: A000027
col 1: A000217
diag (1,5,...): A000326 (pentagonal numbers)
diag (2,7,...): A005449 (second pentagonal numbers)
diag (3,9,...): A045943 (triangular matchstick numbers)
diag (4,11,...): A115067
diag (5,13,...): A140090
diag (6,15,...): A140091
diag (7,17,...): A059845
diag (8,19,...): A140672
antidiagonal sums: A000330
[From Clark Kimberling, ck6(AT)evansville.edu, Feb 5 2011]
(End)
Let N=2*n+1 and k=1,2,...,n. Let A_{N,n-1} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1], an n X n unit-primitive matrix (see [Jeffery]). Let M_n=[A_{N,n-1}]^4. Then t(n,k)=[M_n]_(1,k), that is, the n-th row of the triangle below is given by the first row of M_n. - L. Edson Jeffery, Nov 20 2011
Again, let N=2*n+1 and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n-1} be the n X n unit-primitive matrices (again see [Jeffery]) associated with N, and define the Chebyshev polynomials of the second kind by the recurrence U_0(x)=1, U_1(x)=2*x and U_r(x)=2*x*U_(r-1)(x)-U_(r-2)(x) (r>1). Define the column vectors V_(k-1)=(U_(k-1)(cos(Pi/N)), U_(k-1)(cos(3*Pi/N)), ..., U_(k-1)(cos((2*n-1)*Pi/N)))^T, where B^T denotes the transpose of matrix B. Let S_N=[V_0,V_1,...,V_(n-1)] be the n X n matrix formed by taking the components of vector V_(k-1) as the entries in column k-1 (V_(k-1) gives the ordered spectrum of A_{N,k-1}). Let X_N=[S_N]^T*S_N, and let [X_N]_(i,j) denote the entry in row i and column j of X_N, i,j in {0,...,n-1}. Then also t(n,k)=[X_N]_(k-1,k-1); that is, row n of the triangle is given by the main diagonal entries of X_N. Hence t(n,k) is the sum of squares t(n,k) = sum[m=1,...,n (U_(k-1)(cos((2*m-1)*Pi/N)))^2]. Finally, this sequence is related to A057059, since X_N=[sum_{m=1,...,n} A057059(n,m)*A_{N,m-1}]; i.e., X_N is also an integral linear combination of unit-primitive matrices from the N-th set. - L. Edson Jeffery Jan 20 2012
|
|
|
REFERENCES
| R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.
|
|
|
LINKS
| L. E. Jeffery, Unit-primitive matrices
|
|
|
FORMULA
| t(n,m)=m*(2*n - m + 1)/2.
|
|
|
EXAMPLE
| {1},
{2, 3},
{3, 5, 6},
{4, 7, 9, 10},
{5, 9, 12, 14, 15},
{6, 11, 15, 18, 20, 21},
{7, 13, 18, 22, 25, 27, 28},
{8, 15, 21, 26, 30, 33, 35, 36},
{9, 17, 24, 30, 35, 39, 42, 44, 45},
{10, 19, 27, 34, 40, 45, 49, 52, 54, 55}
As a rectangle:
1....2....3....4....5....6
3....5....7....9....11...13
6....9....12...15...18...21
10...14...18...22...26...30
|
|
|
MATHEMATICA
| Clear[T, n, m, a] T[n_, m_] = m*(2*n - m + 1)/2; a = Table[Table[T[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[a]
|
|
|
CROSSREFS
| Cf. A144112, A051340, A141419, A185874, A185875, A185876.
Cf. A057059.
Sequence in context: A085312 A046530 A003558 * A072451 A023156 A051599
Adjacent sequences: A141416 A141417 A141418 * A141420 A141421 A141422
|
|
|
KEYWORD
| nonn,tabl,changed
|
|
|
AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 05 2008
|
| |
|
|
