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A097862 Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and height k (n>=0, k>=0). 3

%I

%S 1,1,1,1,1,3,1,7,1,1,15,5,1,31,18,1,1,63,56,7,1,127,161,33,1,1,255,

%T 441,129,9,1,511,1170,453,52,1,1,1023,3036,1485,242,11,1,2047,7753,

%U 4644,990,75,1,1,4095,19565,14040,3718,403,13,1,8191,48930,41392,13145,1872,102

%N Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and height k (n>=0, k>=0).

%C Row sums are the Motzkin numbers (A001006).

%H Alois P. Heinz, <a href="/A097862/b097862.txt">Rows n = 0..200, flattened</a>

%F The g.f. for column k is z^(2k)/[P_k*P_{k+1}], where the polynomials P_k are defined by P_0=1, P_1=1-z, P_k=(1-z)P_{k-1}-z^2*P_{k-2}.

%e Triangle begins:

%e 1;

%e 1;

%e 1,1;

%e 1,3;

%e 1,7,1;

%e 1,15,5;

%e 1,31,18,1;

%e 1,63,56,7;

%e 1,127,161,33,1;

%e Row n contains 1+floor(n/2) terms.

%e T(5,2)=5 counts HUUDD, UUDDH, UUDHD, UHUDD and UUHDD, where U=(1,1), H=(1,0) and D=(1,-1).

%p P[0]:=1: P[1]:=1-z: for n from 2 to 10 do P[n]:=sort(expand((1-z)*P[n-1]-z^2*P[n-2])) od: for k from 0 to 8 do h[k]:=series(z^(2*k)/P[k]/P[k+1],z=0,20) od: a:=proc(n,k) if k=0 then 1 elif n=0 then 0 else coeff(h[k],z^n) fi end: seq(seq(a(n,k),k=0..floor(n/2)),n=0..15);

%p # second Maple program:

%p b:= proc(x, y, m) option remember; `if`(y>x, 0,

%p `if`(x=0, z^m, `if`(y>0, b(x-1, y-1, m), 0)+

%p b(x-1, y, m)+b(x-1, y+1, max(m, y+1))))

%p end:

%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):

%p seq(T(n), n=0..16); # _Alois P. Heinz_, Mar 13 2017

%t b[x_, y_, m_] := b[x, y, m] = If[y > x, 0, If[x == 0, z^m, If[y > 0, b[x - 1, y - 1, m], 0] + b[x - 1, y, m] + b[x - 1, y + 1, Max[m, y + 1]]]];

%t T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, 0, 0]];

%t Table[T[n], {n, 0, 16}] // Flatten (* _Jean-Fran├žois Alcover_, May 12 2017, after _Alois P. Heinz_ *)

%Y Cf. A001006.

%K nonn,tabf

%O 0,6

%A _Emeric Deutsch_, Sep 01 2004

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Last modified October 16 08:24 EDT 2018. Contains 316259 sequences. (Running on oeis4.)