

A178519


Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k vertices of outdegree 2 that have (two) leaves as their (two) children.


1



1, 1, 1, 1, 4, 1, 11, 3, 32, 10, 98, 33, 1, 309, 114, 6, 998, 402, 30, 3285, 1439, 137, 1, 10981, 5205, 600, 10, 37178, 18976, 2562, 70, 127227, 69610, 10758, 416, 1, 439369, 256626, 44640, 2250, 15, 1529280, 949974, 183594, 11452, 140, 5359314, 3528725
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,5


COMMENTS

Row n has 1 + floor(n/3) entries (n >= 3).
Sum of entries in row n is A000108(n) (Catalan numbers).
T(n,0) = A178520(n).
Sum_{k>=0} k*T(n,k) = binomial(2n5, n2) = A001700(n3).
Statistic suggested by Lou Shapiro.


LINKS

Table of n, a(n) for n=0..47.


FORMULA

G.f. G=G(t,z) satisfies z*G^2  (1  z^3 + tz^3)*G + 1  z^2 + tz^2 = 0.


EXAMPLE

T(2,1)=1 because we have \/ .
Triangle starts:
1;
1;
1, 1;
4, 1;
11, 3;
32, 10;
98, 33, 1;


MAPLE

eq := z*G^2(1z^3+t*z^3)*G+1z^2+t*z^2: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; 1, 1; for n from 3 to 15 do seq(coeff(P[n], t, j), j = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form


CROSSREFS

Cf. A000108, A178520, A001700.
Sequence in context: A111964 A242351 A124324 * A094503 A113897 A158753
Adjacent sequences: A178516 A178517 A178518 * A178520 A178521 A178522


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, May 31 2010


STATUS

approved



