|
%I #3 Mar 30 2012 17:35:59
%S 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,
%T 87,24,5,1239,641,229,63,15,1,3282,1692,606,172,40,6,8777,4512,1620,
%U 470,110,21,1,23665,12136,4370,1284,311,62,7,64261,32887,11874,3523,880,180
%N Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k peaks at height 1.
%C Row sums are the Motzkin numbers (A001006). Column 0 is A089372.
%F G.f.= 2/[1-z+2z^2-2tz^2+sqrt(1-2z-3z^2)].
%e Triangle begins:
%e 1;
%e 1;
%e 1,1;
%e 2,2;
%e 5,3,1;
%e 12,6,3;
%e Row n has 1+floor(n/2) terms.
%e 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),
%e H=(1,0) and D=(1,-1).
%p 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);
%Y Cf. A001006, A089372.
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Aug 30 2004
|
