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A114327
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Table T(n,m) = n - m read by upwards antidiagonals.
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12
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0, 1, -1, 2, 0, -2, 3, 1, -1, -3, 4, 2, 0, -2, -4, 5, 3, 1, -1, -3, -5, 6, 4, 2, 0, -2, -4, -6, 7, 5, 3, 1, -1, -3, -5, -7, 8, 6, 4, 2, 0, -2, -4, -6, -8, 9, 7, 5, 3, 1, -1, -3, -5, -7, -9, 10, 8, 6, 4, 2, 0, -2, -4, -6, -8, -10, 11, 9, 7, 5, 3, 1, -1, -3, -5, -7, -9, -11, 12, 10, 8, 6, 4, 2, 0, -2, -4, -6, -8, -10, -12
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OFFSET
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0,4
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COMMENTS
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If we arrange A000027 as an array with northwest corner
1 2 4 7 17 ...
3 5 8 12 18 ...
6 9 13 18 24 ...
10 14 19 25 32 ...
diagonals can be numbered as follows depending on their distance to the main diagonal:
Diag 0: 1, 5, 13, 25, ...
Diag 1: 2, 8, 18, 32, ...
Diag -1: 3, 9, 19, 33, ...,
then a(n) in the flattened array is the number of the diagonal that contains n+1. (End)
Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in Jordan-Schwinger form (cf. Harter, Klee, Schwinger). Triangle terms T(n,k) = T(2j,j-m) satisfy: (1/2) T(2j,j-m) = <j,m|J_3|j,m> = m. Matrix J_3 is diagonal, so this equality determines the only nonzero entries. - Bradley Klee, Jan 29 2016
For the characteristic polynomial of the n X n matrix M_n (Det(x*1_n - M_n)) with elements M_n(i, j) = i-j see the Michael Somos, Nov 14 2002, comment on A002415. - Wolfdieter Lang, Feb 05 2018
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LINKS
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J. Schwinger, On Angular Momentum, Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.
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FORMULA
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G.f. for the table: Sum_{n, m>=0} T(n,m)*x^n*y^n = (x-y)/((1-x)^2*(1-y)^2).
E.g.f. for the table: Sum_{n, m>=0} T(n,m)x^n/n!*y^m/m! = (x-y)*e^{x+y}.
G.f. as sequence: -(1+x)/(1-x)^2 + (Sum_{j>=0} (2*j+1)*x^(j*(j+1)/2) / (1-x). The sum is related to Jacobi theta functions. - Robert Israel, Jan 29 2016
Triangle t(n, k) = n - 2*k, for n >= 0, k = 0..n. (see the Maple program). - Wolfdieter Lang, Feb 05 2018
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EXAMPLE
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The table T(n, m) begins:
n\m 0 1 2 3 4 5 ...
0: 0 -1 -2 -3 -4 -5 ...
1: 1 0 -1 -2 -3 -4 ...
2: 2 1 0 -1 -2 -3 ...
3: 3 2 1 0 -1 -2 ...
4: 4 3 2 1 0 -1 ...
5: 5 4 3 2 1 0 ...
...
The triangle t(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 0
1: 1 -1
2: 2 0 -2
3: 3 1 -1 -3
4: 4 2 0 -2 -4
5: 5 3 1 -1 -3 -5
6: 6 4 2 0 -2 -4 -6
7: 7 5 3 1 -1 -3 -5 -7
8: 8 6 4 2 0 -2 -4 -6 -8
9: 9 7 5 3 1 -1 -3 -5 -7 -9
10: 10 8 6 4 2 0 -2 -4 -6 -8 -10
... Reformatted and corrected. (End)
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MAPLE
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MATHEMATICA
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max = 12; a025581 = NestList[Prepend[#, First[#]+1]&, {0}, max]; a002262 = Table[Range[0, n], {n, 0, max}]; a114327 = a025581 - a002262 // Flatten (* Jean-François Alcover, Jan 04 2016 *)
Flatten[Table[-2 m, {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)
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PROG
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(Haskell)
a114327 n k = a114327_tabl !! n !! k
a114327_row n = a114327_tabl !! n
a114327_tabl = zipWith (zipWith (-)) a025581_tabl a002262_tabl
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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