

A141387


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


5



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

0,4


COMMENTS

Construct the infinitedimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in JordanSchwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k)=T(2j,j+m) satisfy:(1/4)T(2j,j+m) = <j,mJ_1^2j,m> = <j,mJ_2^2j,m>. Matrices for J_1^2 and J_2^2 are sparse. These diagonal equalities and the offdiagonal 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 (1p) = 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 Twinlose(n) of a game (win OR lose) is a well known function of z and n. The probabiltiy of the gambler winning Pwin(n) is also known, and is equal to (1z)/(1z^n). Twin(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 Twin(n) and found it to be Twin(n) = Pwin(n) * (the sum of T(n,m) * z^m for m=0 to m=n) where T(n,m) is the given by the nth row of this sequence.  Steve Newman, Oct 24 2016


REFERENCES

R. N. Cahn, SemiSimple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0486449998, 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, 5763.
W. Harter, Principles of Symmetry, Dynamics, Spectroscopy, Wiley, 1993, Ch. 5, page 345346.
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*(nm).


EXAMPLE

{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}


MATHEMATICA

Clear[T, n, m, a]; T[n_, m_] = n + 2* m *(m + n); a = Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[a]
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 * (nm)}; /* Michael Somos, May 28 2017 */


CROSSREFS

[0, 0] together with the row sums give A007290.
Cf. A003991, A268759, A094053, A114327.
Sequence in context: A271320 A331279 A151849 * A335678 A134400 A016095
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



