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 A114327 #94 Apr 07 2025 09:51:56
%S A114327 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,
%T A114327 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,
%U A114327 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
%N A114327 Table T(n,m) = n - m read by upwards antidiagonals.
%C A114327 From _Clark Kimberling_, May 31 2011: (Start)
%C A114327 If we arrange A000027 as an array with northwest corner
%C A114327 1 2 4 7 17 ...
%C A114327 3 5 8 12 18 ...
%C A114327 6 9 13 18 24 ...
%C A114327 10 14 19 25 32 ...
%C A114327 diagonals can be numbered as follows depending on their distance to the main diagonal:
%C A114327 Diag 0: 1, 5, 13, 25, ...
%C A114327 Diag 1: 2, 8, 18, 32, ...
%C A114327 Diag -1: 3, 9, 19, 33, ...,
%C A114327 then a(n) in the flattened array is the number of the diagonal that contains n+1. (End)
%C A114327 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
%C A114327 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
%C A114327 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
%H A114327 Reinhard Zumkeller, <a href="/A114327/b114327.txt">Rows n = 0..125 of triangle, flattened</a>
%H A114327 W. Harter, <a href="https://modphys.hosted.uark.edu/markup/PSDS_Info.html">Principles of Symmetry, Dynamics, Spectroscopy</a>, Wiley, 1993, Ch. 5, page 345-346.
%H A114327 B. Klee, <a href="http://demonstrations.wolfram.com/QuantumAngularMomentumMatrices/">Quantum Angular Momentum Matrices</a>, Wolfram Demonstrations Project, 2016.
%H A114327 J. Schwinger, <a href="http://www.ifi.unicamp.br/~cabrera/teaching/paper_schwinger.pdf">On Angular Momentum</a>, Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.
%F A114327 G.f. for the table: Sum_{n, m>=0} T(n,m)*x^n*y^n = (x-y)/((1-x)^2*(1-y)^2).
%F A114327 E.g.f. for the table: Sum_{n, m>=0} T(n,m)x^n/n!*y^m/m! = (x-y)*e^{x+y}.
%F A114327 T(n,k) = A025581(n,k) - A002262(n,k).
%F A114327 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
%F A114327 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
%F A114327 Triangle t(n, k) = n - 2*k, for n >= 0, k = 0..n. (see the Maple program). - _Wolfdieter Lang_, Feb 05 2018
%e A114327 From _Wolfdieter Lang_, Feb 05 2018: (Start)
%e A114327 The table T(n, m) begins:
%e A114327 n\m 0 1 2 3 4 5 ...
%e A114327 0: 0 -1 -2 -3 -4 -5 ...
%e A114327 1: 1 0 -1 -2 -3 -4 ...
%e A114327 2: 2 1 0 -1 -2 -3 ...
%e A114327 3: 3 2 1 0 -1 -2 ...
%e A114327 4: 4 3 2 1 0 -1 ...
%e A114327 5: 5 4 3 2 1 0 ...
%e A114327 ...
%e A114327 The triangle t(n, k) begins:
%e A114327 n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e A114327 0: 0
%e A114327 1: 1 -1
%e A114327 2: 2 0 -2
%e A114327 3: 3 1 -1 -3
%e A114327 4: 4 2 0 -2 -4
%e A114327 5: 5 3 1 -1 -3 -5
%e A114327 6: 6 4 2 0 -2 -4 -6
%e A114327 7: 7 5 3 1 -1 -3 -5 -7
%e A114327 8: 8 6 4 2 0 -2 -4 -6 -8
%e A114327 9: 9 7 5 3 1 -1 -3 -5 -7 -9
%e A114327 10: 10 8 6 4 2 0 -2 -4 -6 -8 -10
%e A114327 ... Reformatted and corrected. (End)
%p A114327 seq(seq(i-2*j,j=0..i),i=0..30); # _Robert Israel_, Jan 29 2016
%t A114327 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 *)
%t A114327 Flatten[Table[-2 m, {j, 0, 10, 1/2}, {m, -j, j}]] (* _Bradley Klee_, Jan 29 2016 *)
%o A114327 (Haskell)
%o A114327 a114327 n k = a114327_tabl !! n !! k
%o A114327 a114327_row n = a114327_tabl !! n
%o A114327 a114327_tabl = zipWith (zipWith (-)) a025581_tabl a002262_tabl
%o A114327 -- _Reinhard Zumkeller_, Aug 09 2014
%o A114327 (PARI) T(n,m) = n-m \\ _Charles R Greathouse IV_, Feb 07 2017
%o A114327 (Python)
%o A114327 from math import isqrt
%o A114327 def A114327(n): return ((m:=isqrt(k:=n+1<<1))+(k>m*(m+1)))**2+1-k # _Chai Wah Wu_, Nov 09 2024
%Y A114327 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.
%K A114327 easy,sign,tabl,nice
%O A114327 0,4
%A A114327 _Franklin T. Adams-Watters_, Feb 06 2006
%E A114327 Formula improved by _Reinhard Zumkeller_, Aug 09 2014