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A141387
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Triangle read by rows: T(n,m) = n + 2*m*(n - m) (0 <= m <= n).
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7
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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
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OFFSET
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0,4
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COMMENTS
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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
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REFERENCES
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R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.
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LINKS
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J. Schwinger, On Angular Momentum, Cambridge: Harvard University, Nuclear Development Associates, Inc., 1952.
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FORMULA
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T(n,m) = n + 2*m*(n-m).
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EXAMPLE
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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}
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)
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MATHEMATICA
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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 *)
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PROG
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(PARI) {T(n, m) = if( m<0 || n<m, 0, n + 2*m * (n-m)}; /* Michael Somos, May 28 2017 */
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CROSSREFS
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[0, 0] together with the row sums give A007290.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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