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A030237 Catalan's triangle with right border removed. 23
1, 1, 2, 1, 3, 5, 1, 4, 9, 14, 1, 5, 14, 28, 42, 1, 6, 20, 48, 90, 132, 1, 7, 27, 75, 165, 297, 429, 1, 8, 35, 110, 275, 572, 1001, 1430, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This triangle appears in the totally asymmetric exclusion process as Y(alpha=1,beta=1,n,m), written in the Derrida et al. reference as Y_n(m) for alpha=1, beta=1. - Wolfdieter Lang, Jan 13 2006.

REFERENCES

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.

LINKS

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

W. Lang: First 10 rows.

Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757, 2015. See Fig. 6.

FORMULA

T(n,m) = (n-m+1)*binomial(n+m,m)/(n+1).

EXAMPLE

1;

1,2;

1,3,5;

1,4,9,14;

1,5,14,28,42;

1,6,20,48,90,132;

1,7,27,75,165,297,429;

1,8,35,110,275,572,1001,1430;

1,9,44,154,429,1001,2002,3432,4862;

MAPLE

A030237 := proc(n, m)

    (n-m+1)*binomial(n+m, m)/(n+1) ;

end proc: # R. J. Mathar, May 31 2016

MATHEMATICA

T[n_, k_] := T[n, k] = Which[k==0, 1, k>n, 0, True, T[n-1, k] + T[n, k-1]];

Table[T[n, k], {n, 1, 9}, {k, 0, n-1}] // Flatten (* Jean-Fran├žois Alcover, Nov 14 2017 *)

PROG

(Haskell)

a030237 n k = a030237_tabl !! n !! k

a030237_row n = a030237_tabl !! n

a030237_tabl = map init $ tail a009766_tabl

-- Reinhard Zumkeller, Jul 12 2012

CROSSREFS

Cf. A009766.

Row sums give A071724(n).

The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

Sequence in context: A188211 A175009 A049069 * A210557 A118243 A210233

Adjacent sequences:  A030234 A030235 A030236 * A030238 A030239 A030240

KEYWORD

nonn,tabl,easy

AUTHOR

Wouter Meeussen.

EXTENSIONS

Missing a(8) = T(7,0) = 1 inserted by Reinhard Zumkeller, Jul 12 2012

STATUS

approved

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Last modified December 11 05:41 EST 2017. Contains 295868 sequences.