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%I #68 Oct 08 2022 22:20:24
%S 0,1,1,2,4,2,3,7,7,3,4,10,12,10,4,5,13,17,17,13,5,6,16,22,24,22,16,6,
%T 7,19,27,31,31,27,19,7,8,22,32,38,40,38,32,22,8,9,25,37,45,49,49,45,
%U 37,25,9,10,28,42,52,58,60,58,52,42,28,10
%N Triangle read by rows: T(n,m) = n + 2*m*(n - m) (0 <= m <= n).
%C Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k)=T(2j,j+m) satisfy:(1/4)T(2j,j+m) = <j,m|J_1^2|j,m> = <j,m|J_2^2|j,m>. Matrices for J_1^2 and J_2^2 are sparse. These diagonal equalities and the off-diagonal equalities of A268759 determine the only nonzero entries. Comments on A268759 provide a conjecture for the clear interpretation of these numbers in the context of binomial coefficients and other geometrical sequences. - _Bradley Klee_, Feb 20 2016
%C This sequence appears in the probability of the coin tossing "Gambler's Ruin". Call the probability of winning a coin toss = p, and the probability of losing the toss is 1-p = q, and call z = q/p. A gambler starts with $1, and tosses for $1 stakes till he has $0 (ruin) or has $n (wins). The average time T_win_lose(n) of a game (win OR lose) is a well-known function of z and n. The probability of the gambler winning P_win(n) is also known, and is equal to (1-z)/(1-z^n). T_win(n) defined as the average time it takes the gambler to win such a game is not so well known (I have not found it in the literature). I calculated T_win(n) and found it to be T_win(n) = P_win(n) * Sum_{m=0..n} T(n,m) * z^m. - _Steve Newman_, Oct 24 2016
%C As a square array A(n,m), gives the odd number's index of the product of n-th and m-th odd number. See formula. - _Rainer Rosenthal_, Sep 07 2022
%D R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.
%H Michael De Vlieger, <a href="/A141387/b141387.txt">Table of n, a(n) for n = 0..10010</a> (rows 0 <= n <= 70, flattened)
%H Tomislav Došlić, <a href="http://ijmc.kashanu.ac.ir/article_105580.html">On the Laplacian Szeged Spectrum of Paths</a>, Iranian J. Math. Chem. (2020) Vol. 11, No. 1, 57-63.
%H W. Harter, <a href="http://www.uark.edu/ua/modphys/markup/PSDS_Info.html/">Principles of Symmetry, Dynamics, Spectroscopy</a>, Wiley, 1993, Ch. 5, page 345-346.
%H B. Klee, <a href="http://demonstrations.wolfram.com/QuantumAngularMomentumMatrices/">Quantum Angular Momentum Matrices</a>, Wolfram Demonstrations Project, 2016.
%H 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 T(n,m) = n + 2*m*(n-m).
%F Square array A(n,m) = 2*n*m + n + m, read by antidiagonals, satisfying 2*A(n,m) + 1 = (2*n+1)*(2*m+1) = A005408(n)*A005408(m) = A098353(n+1,m+1). - _Rainer Rosenthal_, Oct 01 2022
%e As a triangle:
%e { 0},
%e { 1, 1},
%e { 2, 4, 2},
%e { 3, 7, 7, 3},
%e { 4, 10, 12, 10, 4},
%e { 5, 13, 17, 17, 13, 5},
%e { 6, 16, 22, 24, 22, 16, 6},
%e { 7, 19, 27, 31, 31, 27, 19, 7},
%e { 8, 22, 32, 38, 40, 38, 32, 22, 8},
%e { 9, 25, 37, 45, 49, 49, 45, 37, 25, 9},
%e {10, 28, 42, 52, 58, 60, 58, 52, 42, 28, 10}
%e From _Peter Munn_, Sep 28 2022: (Start)
%e Square array A(n,m) starts:
%e 0, 1, 2, 3, 4, 5, 6, 7, ...
%e 1, 4, 7, 10, 13, 16, 19, 22, ...
%e 2, 7, 12, 17, 22, 27, 32, 37, ...
%e 3, 10, 17, 24, 31, 38, 45, 52, ...
%e 4, 13, 22, 31, 40, 49, 58, 67, ...
%e 5, 16, 27, 38, 49, 60, 71, 82, ...
%e 6, 19, 32, 45, 58, 71, 84, 97, ...
%e ...
%e (End)
%t T[n_, m_] = n + 2* m *(-m + n);
%t a = Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}];
%t Flatten[a]
%t (* second program: *)
%t Flatten[ Table[2 j + 2 j^2 - 2 m^2, {j, 0, 10, 1/2}, {m, -j, j}]] (* _Bradley Klee_, Feb 20 2016 *)
%o (PARI) {T(n, m) = if( m<0 || n<m, 0, n + 2*m * (n-m)}; /* _Michael Somos_, May 28 2017 */
%Y [0, 0] together with the row sums give A007290.
%Y Cf. A003991, A268759, A094053, A098353, A114327.
%K nonn,tabl
%O 0,4
%A _Roger L. Bagula_, Aug 03 2008
%E Edited by _N. J. A. Sloane_, Feb 21 2016
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