OFFSET
1,4
COMMENTS
T(n,k) = A003991(n-1,k) for 1 <= k < n;
T(n,k) = T(n,n-1-k) for k < n;
T(n,1) = n-1; T(n,n) = 0; T(n,2) = A005843(n-2) for n > 1;
All numbers m > 0 occur A000005(m) times;
k-th columns are arithmetic progressions with step k, starting with 0. If a zero is prefixed to the sequence, then we get a new table where the columns are again arithmetic progressions with step k, but starting with k, k=0,1,2,...: 1st column = (0,0,0,...), 2nd column = (1,2,3,...), 3rd column = (2,4,6,8,...), etc. - M. F. Hasler, Feb 02 2013
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) = T(2j,j+m) satisfy: (1/2)T(2j,j+m)^(1/2) = <j,m|J_1|j,m+1> = <j,m+1|J_1|j,m> = i <j,m|J_2|j,m+1> = -i <j,m+1|J_2|j,m>. Matrices for J_1 and J_2 are sparse. These equalities determine the only nonzero entries. - Bradley Klee, Jan 29 2016
T(n+1,k+1) is the number of degrees of freedom of a k-dimensional affine subspace within an n-dimensional vector space. This is most readily interpreted geometrically: e.g. in 3 dimensions a line (1-dimensional subspace) has T(4,2) = 4 degrees of freedom and a plane has T(4,3) = 3. T(n+1,1) = n indicates that points in n dimensions have n degrees of freedom. T(n,n) = 0 for any n as all n-dimensional spaces in an n-dimensional space are equivalent. - Daniel Leary, Apr 29 2020
LINKS
W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345-346.
B. Klee, Quantum Angular Momentum Matrices, Wolfram Demonstrations Project, 2016.
J. Schwinger, On Angular Momentum , Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.
EXAMPLE
From M. F. Hasler, Feb 02 2013: (Start)
Triangle begins:
0;
1, 0;
2, 2, 0;
3, 4, 3, 0;
4, 6, 6, 4, 0;
5, 8, 9, 8, 5, 0;
(...)
If an additional 0 was added at the beginning, this would become:
0;
0, 1;
0, 2, 2;
0, 3, 4; 3;
0, 4, 6, 6, 4;
0, 5, 8, 9, 8, 5;
... (End)
MATHEMATICA
Flatten[Table[(j - m) (j + m + 1), {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)
PROG
(Magma) /* As triangle */ [[k*(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jan 30 2016
(PARI) {for(n=1, 13, for(k=1, n, print1(k*(n - k), " "); ); print(); ); } \\ Indranil Ghosh, Mar 12 2017
CROSSREFS
T(n,k) for values of k:
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, May 31 2004
STATUS
approved