2,3

Basically, the mirror image of A020474. Row n has floor(n/2) terms (first row is row 2). Row sums yield the Riordan numbers (A005043). Column 1 yields the Motzkin numbers (A001006); column 2 yields A002026; column 3 yields A005322; column 4 yields A005323; column 4 yields A005324; column 5 yields A005325; column 6 yields A005326.

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

G.f.=tz^2*S/(1-zS-tz^2*S), where S=S(z)=[1+z-sqrt(1-2z-3z^2)]/[2z(1+z)] is the g.f. of the short bushes (the Riordan numbers; A005043).

Column 1 yields the Motzkin numbers: indeed, if from each short bush, having leftmost leaf at height 1, we drop the leftmost edge, then we obtain the

so-called bushes, known to be counted by the Motzkin numbers.

Triangle begins:

1;

1;

2,1;

4,2;

9,5,1;

21,12,3;

51,30,9,1.

S:=1/2/(z+z^2)*(1+z-sqrt(1-2*z-3*z^2)): G:=simplify(t*z^2*S/(1-z*S-t*z^2*S)): Gserz:=simplify(series(G, z=0, 19)): for n from 2 to 17 do P[n]:=sort(coeff(Gserz, z^n)) od: for n from 2 to 17 do seq(coeff(P[n], t^k), k=1..floor(n/2)) od; # yields sequence in triangular form

Cf. A020474, A005043, A001006, A002026, A005322, A005323, A005324, A005325, A005326.

Sequence in context: A135530 A137206 A076736 this_sequence A132280 A059970 A112157

Adjacent sequences: A106486 A106487 A106488 this_sequence A106490 A106491 A106492

nonn,tabf

Emeric Deutsch (deutsch(AT)duke.poly.edu), May 29 2005