A114327 Table T(n,m) = n - m read by upwards antidiagonals.

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

Offset: 0

Author

Franklin T. Adams-Watters, Feb 06 2006

Keywords

• easy
• sign
• tabl
• nice

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) = = 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

The entries of the n-th antidiagonal, T(n,1), T(n-1,2), ... , T(1,n), are the eigenvalues of the Hamming graph H(2,n-1) (or hypercube Q(n-1)). - Miquel A. Fiol, May 21 2024

Examples

			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)
		

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.

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.

Programs

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

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-François Alcover, Jan 04 2016 *)
Flatten[Table[-2 m, {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Jan 29 2016 *)

PARI

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

Python

from math import isqrt
def A114327(n): return ((m:=isqrt(k:=n+1<<1))+(k>m*(m+1)))**2+1-k # Chai Wah Wu, Nov 09 2024

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+1) = A004736(n) - A002260(n) or a(n+1) = ((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

Extensions

Formula improved by Reinhard Zumkeller, Aug 09 2014