OFFSET
1,2
COMMENTS
A (totally) symmetric traceless tensor of rank r >= 2 and dimension n >= 1 is irreducible.
The array of the number of independent components of a rank r symmetric traceless tensor A(r, n), for r >= 2 and n >=1, is given by risefac(n,r)/r! - risefac(n,r-2)/(r-2)!, where the first term gives the number of independent components of a symmetric tensors of rank r (see a Dec 10 2015 comment under A135278) and the second term is the number of constraints from the tracelessness requirement. The tensor has to be traceless in each pair of indices.
The first rows of the array A, or the first columns (without the first r-2 zeros) of the triangle T are for r = 2..6: A000096, A005581, A005582, A005583, A005584.
Equals A115241 with the first column of positive integers removed. - Georg Fischer, Jul 26 2023
FORMULA
T(n, r) = A(r, n-r+2) with the array A(r, n) = risefac(n,r)/r! - risefac(n,r-2)/(r-2)! where the rising factorial risefac(n,k) = Product_{j=0..k-1} (n+j) and risefac(n,0) = 1.
From Peter Luschny, Dec 14 2015: (Start)
A(n+2, n+1) = A007946(n-1) = CatalanNumber(n)*3*n*(n+1)/(n+2) for n>=0.
A(n+2, n+3) = A051960(n+1) = CatalanNumber(n+1)*(3*n+5) for n>=0.
A(n+2, n+4) = A029651(n+2) = CatalanNumber(n+1)*(6*n+9) for n>=0.
A(n+2, n+5) = A051924(n+3) = CatalanNumber(n+2)*(3*n+7) for n>=0.
A(n+2, n+6) = A129869(n+4) = CatalanNumber(n+2)*(3*n+8)*(2*n+5)/(n+4) for n>=0.
A(n+2, n+7) = A220101(n+4) = CatalanNumber(n+3)*(3*(n+3)^2)/(n+5) for n>=0.
A(n+2, n+8) = CatalanNumber(n+4)*(n+3)*(3*n+10)/(2*n+12) for n>=0.
Let for n>=0 and k>=0 diag(n,k) = A(k+2,n+k+1) and G(n,k) = 2^(k+2*n)*Gamma((3-(-1)^k+2*k+4*n)/4)/(sqrt(Pi)*Gamma(k+n+0^k)) then
diag(n,0) = G(n,0)*(n*3)/(n+2),
diag(n,1) = G(n,1)*(3*n+4)/((n+1)*(n+2)),
diag(n,2) = G(n,2)*(3*n+5)/(n+2),
diag(n,3) = G(n,3)*3,
diag(n,4) = G(n,4)*(3*n+7),
diag(n,5) = G(n,5)*(3*n+8),
diag(n,6) = G(n,6)*3*(3+n)^2,
diag(n,7) = G(n,7)*(3+n)*(10+3*n). (End)
EXAMPLE
The array A(r, n) starts:
r\n 1 2 3 4 5 6 7 8 9 10 ...
2: 0 2 5 9 14 20 27 35 44 54
3: 0 2 7 16 30 50 77 112 156 210
4: 0 2 9 25 55 105 182 294 450 660
5: 0 2 11 36 91 196 378 672 1122 1782
6: 0 2 13 49 140 336 714 1386 2508 4290
7: 0 2 15 64 204 540 1254 2640 5148 9438
8: 0 2 17 81 285 825 2079 4719 9867 19305
9: 0 2 19 100 385 1210 3289 8008 17875 37180
10: 0 2 21 121 506 1716 5005 13013 30888 68068
...
The triangle T(n, r) starts:
n\r 2 3 4 5 6 7 8 9 10 11 ...
1: 0
2: 2 0
3: 5 2 0
4: 9 7 2 0
5: 14 16 9 2 0
6: 20 30 25 11 2 0
7: 27 50 55 36 13 2 0
8: 35 77 105 91 49 15 2 0
9: 44 112 182 196 140 64 17 2 0
10: 54 156 294 378 336 204 81 19 2 0
...
A(r, 1) = 0 , r >= 2, because a symmetric rank r tensor t of dimension one has one component t(1,1,...,1) (r 1's) and if the traces vanish then t vanishes.
A(3, 2) = 2 because a symmetric rank 3 tensor t with three indices taking values from 1 or 2 (n=2) has the four independent components t(1,1,1), t(1,1,2), t(1,2,2), t(2,2,2), and (invoking symmetry) the vanishing traces are Sum_{j=1..2} t(j,j,1) = 0 and Sum_{j=1..2} t(j,j,2) = 0. These are two constraints, which can be used to eliminate, say, t(1,1,1) and t(2,2,2), leaving 2 = A(3, 2) independent components, say, t(1,1,2) and t(1,2,2).
From Peter Luschny, Dec 14 2015: (Start)
The diagonals diag(n, k) start:
k\n 0 1 2 3 4 5 6
0: 0, 2, 9, 36, 140, 540, 2079, ... A007946
1: 2, 7, 25, 91, 336, 1254, 4719, ... A097613
2: 5, 16, 55, 196, 714, 2640, 9867, ... A051960
3: 9, 30, 105, 378, 1386, 5148, 19305, ... A029651
4: 14, 50, 182, 672, 2508, 9438, 35750, ... A051924
5: 20, 77, 294, 1122, 4290, 16445, 63206, ... A129869
6: 27, 112, 450, 1782, 7007, 27456, 107406, ... A220101
7: 35, 156, 660, 2717, 11011, 44200, 176358, ... A265612
8: 44, 210, 935, 4004, 16744, 68952, 281010, ... A265613
(End)
MATHEMATICA
A[r_, n_] := Pochhammer[n, r]/r! - Pochhammer[n, r-2]/(r-2)!;
T[n_, r_] := A[r, n-r+2];
Table[T[n, r], {n, 1, 10}, {r, 2, n+1}] (* Jean-François Alcover, Jun 28 2019 *)
PROG
(Sage)
A = lambda r, n: rising_factorial(n, r)/factorial(r) - rising_factorial(n, r-2)/factorial(r-2)
for r in (2..10): [A(r, n) for n in (1..10)] # Peter Luschny, Dec 13 2015
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Dec 10 2015
STATUS
approved