login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A026300 Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1). 39
1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 12, 9, 1, 5, 14, 25, 30, 21, 1, 6, 20, 44, 69, 76, 51, 1, 7, 27, 70, 133, 189, 196, 127, 1, 8, 35, 104, 230, 392, 518, 512, 323, 1, 9, 44, 147, 369, 726, 1140, 1422, 1353, 835, 1, 10, 54, 200, 560, 1242, 2235, 3288, 3915, 3610, 2188 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Right-hand columns have g.f. M^k, where M is g.f. of Motzkin numbers.

Consider a semi-infinite chessboard with squares labeled (n,k), ranks or rows n >= 0, files or columns k >= 0; number of king-paths of length n from (0,0) to (n,k), 0 <= k <= n, is T(n,n-k). - Harrie Grondijs, May 27 2005. Cf. A114929, A111808, A114972.

REFERENCES

Xiang-Ke Chang, XB Hu, H Lei, YN Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.

Harrie Grondijs, Neverending Quest of Type C, Volume B - the endgame study-as-struggle.

A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.

LINKS

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675.

Tewodros Amdeberhan, Moa Apagodu, Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015.

J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

H. Bottomley, Illustration of initial terms

M. Buckley, R. Garner, S. Lack, R. Street, Skew-monoidal categories and the Catalan simplicial set, arXiv preprint arXiv:1307.0265 [math.CT], 2013.

L. Carlitz, Solution of certain recurrences, SIAM J. Appl. Math., 17 (1969), 251-259.

J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.

I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv preprint arXiv:1507.04838, 2015

A. Luzón, D. Merlini, M. A. Morón, R. Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.

A. Roshan, P. H. Jones and C. D. Greenman, An Exact, Time-Independent Approach to Clone Size Distributions in Normal and Mutated Cells, arXiv preprint arXiv:1311.5769 [q-bio.QM], 2013.

Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, PDF.

Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, Combinatorics, Probability and Computing, Volume 24, Special Issue 01,January 2015, pp 354-372.

FORMULA

T(n,k) = Sum_{i=0..floor(k/2)} binomial(n, 2i+n-k)(binomial(2i+n-k, i) - binomial(2i+n-k, i-1)). - Herbert Kociemba, May 27 2004

T(n,k) = A027907(n,k) - A027907(n,k-2), k<=n.

Sum_{k=0..n}(-1)^k*T(n,k) = A099323(n+1). - Philippe Deléham, Mar 19 2007

Sum_{k=0..n}(T(n,k) mod 2) = A097357(n+1). - Philippe Deléham, Apr 28 2007

Sum_{k=0..n} T(n,k)*x^(n-k) = A005043(n), A001006(n), A005773(n+1), A059738(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Nov 28 2009

EXAMPLE

1

1 1

1 2  2

1 3  5  4

1 4  9 12  9

1 5 14 25 30 21

...

MAPLE

A026300 := proc(n, k)

    add(binomial(n, 2*i+n-k)*(binomial(2*i+n-k, i) -binomial(2*i+n-k, i-1)) , i=0..floor(k/2)) ;

end proc: # R. J. Mathar, Jun 30 2013

MATHEMATICA

t[n_, k_] := Sum[ Binomial[n, 2i + n - k] (Binomial[2i + n - k, i] - Binomial[2i + n - k, i - 1]), {i, 0, Floor[k/2]}]; Table[ t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 03 2011 *)

t[_, 0] = 1; t[n_, 1] := n; t[n_, k_] /; k>n || k<0 = 0; t[n_, n_] := t[n, n] = t[n-1, n-2]+t[n-1, n-1]; t[n_, k_] := t[n, k] = t[n-1, k-2]+t[n-1, k-1]+t[n-1, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 18 2014 *)

PROG

(Haskell)

a026300 n k = a026300_tabl !! n !! k

a026300_row n = a026300_tabl !! n

a026300_tabl = iterate (\row -> zipWith (+) ([0, 0] ++ row) $

                                zipWith (+) ([0] ++ row) (row ++ [0])) [1]

-- Reinhard Zumkeller, Oct 09 2013

(PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(sum(i=0, k\2, binomial(n, 2*i+n-k)*(binomial(2*i+n-k, i)-binomial(2*i+n-k, i-1))), ", "); ); print(); ); } \\ Michel Marcus, Jul 25 2015

CROSSREFS

Motzkin numbers (A001006) are T(n,n), other columns of T include A002026, A005322, A005323.

Cf. A020474.

Row sums are in A005773.

Reflected version is in A064189. Cf. A059738.

Sequence in context: A140717 A257005 A160232 * A099514 A228352 A205575

Adjacent sequences:  A026297 A026298 A026299 * A026301 A026302 A026303

KEYWORD

nonn,tabl,nice

AUTHOR

Clark Kimberling

EXTENSIONS

Corrected and edited by Johannes W. Meijer, Oct 05 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 21:48 EDT 2017. Contains 284288 sequences.