A141387 Triangle read by rows: T(n,m) = n + 2*m*(n - m) (0 <= m <= n).

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, 7, 19, 27, 31, 31, 27, 19, 7, 8, 22, 32, 38, 40, 38, 32, 22, 8, 9, 25, 37, 45, 49, 49, 45, 37, 25, 9, 10, 28, 42, 52, 58, 60, 58, 52, 42, 28, 10

Offset

0,4

Comments

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

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

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

References

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10010 (rows 0 <= n <= 70, flattened)

Tomislav Došlić, On the Laplacian Szeged Spectrum of Paths, Iranian J. Math. Chem. (2020) Vol. 11, No. 1, 57-63.

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

T(n,m) = n + 2*m*(n-m).

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

Example

As a triangle:
  { 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},
  { 7, 19, 27, 31, 31, 27, 19,  7},
  { 8, 22, 32, 38, 40, 38, 32, 22,  8},
  { 9, 25, 37, 45, 49, 49, 45, 37, 25,  9},
  {10, 28, 42, 52, 58, 60, 58, 52, 42, 28, 10}
From Peter Munn, Sep 28 2022: (Start)
Square array A(n,m) starts:
  0,  1,  2,  3,  4,  5,  6,  7, ...
  1,  4,  7, 10, 13, 16, 19, 22, ...
  2,  7, 12, 17, 22, 27, 32, 37, ...
  3, 10, 17, 24, 31, 38, 45, 52, ...
  4, 13, 22, 31, 40, 49, 58, 67, ...
  5, 16, 27, 38, 49, 60, 71, 82, ...
  6, 19, 32, 45, 58, 71, 84, 97, ...
...
(End)

Mathematica

T[n_, m_] = n + 2* m *(-m + n);
a = Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[a]
(* second program: *)
Flatten[ Table[2 j + 2 j^2 - 2 m^2, {j, 0, 10, 1/2}, {m, -j, j}]] (* Bradley Klee, Feb 20 2016 *)

Prog

(PARI) {T(n, m) = if( m<0 || n<m, 0, n + 2*m * (n-m)}; /* Michael Somos, May 28 2017 */

Crossrefs

[0, 0] together with the row sums give A007290.

Cf. A003991, A268759, A094053, A098353, A114327.

Sequence in context: A331279 A151849 A360373 * A349400 A335678 A368434

Adjacent sequences: A141384 A141385 A141386 * A141388 A141389 A141390

Keyword

nonn,tabl

Author

Roger L. Bagula, Aug 03 2008

Extensions

Edited by N. J. A. Sloane, Feb 21 2016

Status

approved