

A097609


Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k horizontal steps at level 0.


11



1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 3, 2, 3, 0, 1, 6, 7, 3, 4, 0, 1, 15, 14, 12, 4, 5, 0, 1, 36, 37, 24, 18, 5, 6, 0, 1, 91, 90, 67, 36, 25, 6, 7, 0, 1, 232, 233, 165, 106, 50, 33, 7, 8, 0, 1, 603, 602, 438, 264, 155, 66, 42, 8, 9, 0, 1, 1585, 1586, 1147, 719, 390, 215, 84, 52, 9, 10, 0, 1
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OFFSET

0,8


COMMENTS

Row sums give the Motzkin numbers (A001006).
Column 0 is A005043.
Riordan array ((1+xsqrt(12x3x^2))/(2x(1x)),(1+xsqrt(12x3x^2))/(2(1x))).  Paul Barry, Jun 21 2008
Inverse of Riordan array ((1x)/(1x+x^2),x(1x)/(1x+x^2)), which is A104597.  Paul Barry, Jun 21 2008
Triangle read by rows, product of A064189 and A130595 considered as infinite lower triangular arrays; A097609 = A064189*A130195 = B*A053121*B^(1) where B = A007318.  Philippe Deléham, Dec 07 2009
T(n+1,1) = A187306(n).  Philippe Deléham, Jan 28 2014
The number of lattice paths from (0,0) to (n,k) that do not cross below the xaxis and use upstep=(1,1) and downsteps=(1,z) where z is a positive integer. For example, T(4,0) = 3: [(1,1)(1,1)(1,1)(1,1)], [(1,1)(1,1)(1,1)(1,1)] and [(1,1)(1,1)(1,1)(1,3)].  Nicholas Ham, Aug 20 2015


LINKS

Table of n, a(n) for n=0..77.
I Dolinka, J East, RD Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279, 2015.
D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees, Trends in Mathematics 2000, pp 127139.


FORMULA

G.f.: 2/(12*t*z+z+sqrt(12*z3*z^2)).
T(n,k) = T(n1,k1)+ Sum_{j, j>=1} T(n1,k+j) ; T(0,0)=1 . [Philippe Deléham, Jan 23 2010]
T(n,k) = k/n*sum(j=k..n, C(n,j)*C(2*jk1,j1)*(1)^(nj)), n>0.  Vladimir Kruchinin, Feb 05 2011


EXAMPLE

Triangle begins:
1;
0,1;
1,0,1;
1,2,0,1;
3,2,3,0,1;
6,7,3,4,0,1;
Row n has n+1 terms.
T(5,2) = 3 because (HH)UHD,(H)UHD(H) and UHD(HH) are the only Motzkin paths of length 5 with 2 horizontal steps at level 0 (shown between parentheses); here U=(1,1), H=(1,0) and D=(1,1).
Production matrix begins
0, 1
1, 0, 1
1, 1, 0, 1
1, 1, 1, 0, 1
1, 1, 1, 1, 0, 1
1, 1, 1, 1, 1, 0, 1
1, 1, 1, 1, 1, 1, 0, 1
1, 1, 1, 1, 1, 1, 1, 0, 1
1, 1, 1, 1, 1, 1, 1, 1, 0, 1
...  Philippe Deléham, Mar 02 2013


MAPLE

G:=2/(12*t*z+z+sqrt(12*z3*z^2)): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser, z^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..n+1), n=0..12);


MATHEMATICA

nmax = 12; t[n_, k_] := ((1)^(n+k)*k*n!*HypergeometricPFQ[{(k+1)/2, k/2, kn}, {k, k+1}, 4])/(n*k!*(nk)!); Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 1, n}]] (* JeanFrançois Alcover, Nov 14 2011, after Vladimir Kruchinin *)


CROSSREFS

Cf. A001006, A005043, A187306.
Sequence in context: A091889 A147785 A067591 * A266692 A077884 A267724
Adjacent sequences: A097606 A097607 A097608 * A097610 A097611 A097612


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Aug 30 2004


STATUS

approved



