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A030237 Catalan's triangle with right border removed (n > 0, 0 <= k < n). 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=1..151 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-François 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

(PARI) T(n, k) = (n-k+1)*binomial(n+k, k)/(n+1) \\ Andrew Howroyd, Feb 23 2018

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: A297395 A297595 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 16 04:22 EST 2019. Contains 330013 sequences. (Running on oeis4.)