A097611 Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k peaks at height 1.

1, 1, 1, 1, 2, 2, 5, 3, 1, 12, 6, 3, 29, 15, 6, 1, 72, 38, 13, 4, 183, 96, 33, 10, 1, 473, 246, 87, 24, 5, 1239, 641, 229, 63, 15, 1, 3282, 1692, 606, 172, 40, 6, 8777, 4512, 1620, 470, 110, 21, 1, 23665, 12136, 4370, 1284, 311, 62, 7, 64261, 32887, 11874, 3523, 880, 180

Offset

0,5

Comments

Row sums are the Motzkin numbers (A001006). Column 0 is A089372.

Links

Table of n, a(n) for n=0..61.

Formula

G.f.= 2/[1-z+2z^2-2tz^2+sqrt(1-2z-3z^2)].

Example

Triangle begins:
1;
1;
1,1;
2,2;
5,3,1;
12,6,3;
Row n has 1+floor(n/2) terms.
T(5,2)=3 because H(UD)(UD), (UD)H(UD), (UD)(UD)H are the only Motzkin paths of length 5 with 2 peaks at height 1 (shown between parentheses); here U=(1,1),
H=(1,0) and D=(1,-1).

Maple

G := 2/(1-z+sqrt(1-2*z-3*z^2)+2*z^2-2*z^2*t): Gser:=simplify(series(G, z=0, 16)): P[0]:=1: for n from 1 to 15 do P[n]:=sort(coeff(Gser, z^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)), n=0..15);

Crossrefs

Cf. A001006, A089372.

Sequence in context: A038041 A197591 A097891 * A135376 A132850 A076561

Adjacent sequences: A097608 A097609 A097610 * A097612 A097613 A097614

Keyword

nonn,tabf

Author

Emeric Deutsch, Aug 30 2004

Status

approved