This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A264357 #40 Jul 26 2023 14:32:45
%S A264357 0,2,0,5,2,0,9,7,2,0,14,16,9,2,0,20,30,25,11,2,0,27,50,55,36,13,2,0,
%T A264357 35,77,105,91,49,15,2,0,44,112,182,196,140,64,17,2,0,54,156,294,378,
%U A264357 336,204,81,19,2,0
%N A264357 Array A(r, n) of number of independent components of a symmetric traceless tensor of rank r and dimension n, written as triangle T(n, r) = A(r, n-r+2), n >= 1, r = 2..n+1.
%C A264357 A (totally) symmetric traceless tensor of rank r >= 2 and dimension n >= 1 is irreducible.
%C A264357 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.
%C A264357 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.
%C A264357 Equals A115241 with the first column of positive integers removed. - _Georg Fischer_, Jul 26 2023
%F A264357 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.
%F A264357 From _Peter Luschny_, Dec 14 2015: (Start)
%F A264357 A(n+2, n+1) = A007946(n-1) = CatalanNumber(n)*3*n*(n+1)/(n+2) for n>=0.
%F A264357 A(n+2, n+2) = A024482(n+2) = A097613(n+2) = CatalanNumber(n+1)*(3*n+4)/2 for n>=0.
%F A264357 A(n+2, n+3) = A051960(n+1) = CatalanNumber(n+1)*(3*n+5) for n>=0.
%F A264357 A(n+2, n+4) = A029651(n+2) = CatalanNumber(n+1)*(6*n+9) for n>=0.
%F A264357 A(n+2, n+5) = A051924(n+3) = CatalanNumber(n+2)*(3*n+7) for n>=0.
%F A264357 A(n+2, n+6) = A129869(n+4) = CatalanNumber(n+2)*(3*n+8)*(2*n+5)/(n+4) for n>=0.
%F A264357 A(n+2, n+7) = A220101(n+4) = CatalanNumber(n+3)*(3*(n+3)^2)/(n+5) for n>=0.
%F A264357 A(n+2, n+8) = CatalanNumber(n+4)*(n+3)*(3*n+10)/(2*n+12) for n>=0.
%F A264357 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
%F A264357 diag(n,0) = G(n,0)*(n*3)/(n+2),
%F A264357 diag(n,1) = G(n,1)*(3*n+4)/((n+1)*(n+2)),
%F A264357 diag(n,2) = G(n,2)*(3*n+5)/(n+2),
%F A264357 diag(n,3) = G(n,3)*3,
%F A264357 diag(n,4) = G(n,4)*(3*n+7),
%F A264357 diag(n,5) = G(n,5)*(3*n+8),
%F A264357 diag(n,6) = G(n,6)*3*(3+n)^2,
%F A264357 diag(n,7) = G(n,7)*(3+n)*(10+3*n). (End)
%e A264357 The array A(r, n) starts:
%e A264357 r\n 1 2 3 4 5 6 7 8 9 10 ...
%e A264357 2: 0 2 5 9 14 20 27 35 44 54
%e A264357 3: 0 2 7 16 30 50 77 112 156 210
%e A264357 4: 0 2 9 25 55 105 182 294 450 660
%e A264357 5: 0 2 11 36 91 196 378 672 1122 1782
%e A264357 6: 0 2 13 49 140 336 714 1386 2508 4290
%e A264357 7: 0 2 15 64 204 540 1254 2640 5148 9438
%e A264357 8: 0 2 17 81 285 825 2079 4719 9867 19305
%e A264357 9: 0 2 19 100 385 1210 3289 8008 17875 37180
%e A264357 10: 0 2 21 121 506 1716 5005 13013 30888 68068
%e A264357 ...
%e A264357 The triangle T(n, r) starts:
%e A264357 n\r 2 3 4 5 6 7 8 9 10 11 ...
%e A264357 1: 0
%e A264357 2: 2 0
%e A264357 3: 5 2 0
%e A264357 4: 9 7 2 0
%e A264357 5: 14 16 9 2 0
%e A264357 6: 20 30 25 11 2 0
%e A264357 7: 27 50 55 36 13 2 0
%e A264357 8: 35 77 105 91 49 15 2 0
%e A264357 9: 44 112 182 196 140 64 17 2 0
%e A264357 10: 54 156 294 378 336 204 81 19 2 0
%e A264357 ...
%e A264357 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.
%e A264357 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).
%e A264357 From _Peter Luschny_, Dec 14 2015: (Start)
%e A264357 The diagonals diag(n, k) start:
%e A264357 k\n 0 1 2 3 4 5 6
%e A264357 0: 0, 2, 9, 36, 140, 540, 2079, ... A007946
%e A264357 1: 2, 7, 25, 91, 336, 1254, 4719, ... A097613
%e A264357 2: 5, 16, 55, 196, 714, 2640, 9867, ... A051960
%e A264357 3: 9, 30, 105, 378, 1386, 5148, 19305, ... A029651
%e A264357 4: 14, 50, 182, 672, 2508, 9438, 35750, ... A051924
%e A264357 5: 20, 77, 294, 1122, 4290, 16445, 63206, ... A129869
%e A264357 6: 27, 112, 450, 1782, 7007, 27456, 107406, ... A220101
%e A264357 7: 35, 156, 660, 2717, 11011, 44200, 176358, ... A265612
%e A264357 8: 44, 210, 935, 4004, 16744, 68952, 281010, ... A265613
%e A264357 A000096,A005581,A005582,A005583,A005584.
%e A264357 (End)
%t A264357 A[r_, n_] := Pochhammer[n, r]/r! - Pochhammer[n, r-2]/(r-2)!;
%t A264357 T[n_, r_] := A[r, n-r+2];
%t A264357 Table[T[n, r], {n, 1, 10}, {r, 2, n+1}] (* _Jean-François Alcover_, Jun 28 2019 *)
%o A264357 (Sage)
%o A264357 A = lambda r, n: rising_factorial(n,r)/factorial(r) - rising_factorial(n,r-2)/factorial(r-2)
%o A264357 for r in (2..10): [A(r,n) for n in (1..10)] # _Peter Luschny_, Dec 13 2015
%Y A264357 Cf. A000096, A005581, A005582, A005583, A005584, A007946, A024482, A029651, A051924, A051960, A097613, A115241, A129869, A135278, A220101, A265612, A265613.
%K A264357 nonn,easy,tabl
%O A264357 1,2
%A A264357 _Wolfdieter Lang_, Dec 10 2015