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A114327 Table T(n,m) = n - m read by upwards antidiagonals. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

From Clark Kimberling, May 31 2011: (Start)

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

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

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.

FORMULA

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}.

T(n,k) = A025581(n,k) - A002262(n,k).

a(n) = A004736(n) - A002260(n) or a(n) = (t*t+3*t+4)/2-n) - (n-t(t+1)/2), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012

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

EXAMPLE

From Wolfdieter Lang, Feb 05 2018: (Start)

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)

MAPLE

seq(seq(i-2*j, j=0..i), i=0..30); # Robert Israel, Jan 29 2016

MATHEMATICA

max = 12; a025581 = NestList[Prepend[#, First[#]+1]&, {0}, max]; a002262 = Table[Range[0, n], {n, 0, max}]; a114327 = a025581 - a002262 // Flatten (* Jean-Fran├žois Alcover, Jan 04 2016 *)

Flatten[Table[-2 m, {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

PROG

(Haskell)

a114327 n k = a114327_tabl !! n !! k

a114327_row n = a114327_tabl !! n

a114327_tabl = zipWith (zipWith (-)) a025581_tabl a002262_tabl

-- Reinhard Zumkeller, Aug 09 2014

(PARI) T(n, m) = n-m \\ Charles R Greathouse IV, Feb 07 2017

CROSSREFS

Apart from signs, same as A049581. Cf. A003056, A025581, A002262, A002260, A004736. J_1,J_2: A094053; J_1^2,J_2^2: A141387, A268759. A002415.

Sequence in context: A257570 A220417 A049581 * A231154 A073450 A284592

Adjacent sequences:  A114324 A114325 A114326 * A114328 A114329 A114330

KEYWORD

easy,sign,tabl,nice

AUTHOR

Franklin T. Adams-Watters, Feb 06 2006

EXTENSIONS

Formula improved by Reinhard Zumkeller, Aug 09 2014

STATUS

approved

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)