

A121685


Triangle read by rows: T(n,k) is the number of binary trees having n edges and k branches (1<=k<=n). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.


1



2, 4, 1, 8, 4, 2, 16, 12, 12, 2, 32, 32, 48, 16, 4, 64, 80, 160, 80, 40, 5, 128, 192, 480, 320, 240, 60, 10, 256, 448, 1344, 1120, 1120, 420, 140, 14, 512, 1024, 3584, 3584, 4480, 2240, 1120, 224, 28, 1024, 2304, 9216, 10752, 16128, 10080, 6720, 2016, 504, 42
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

The row sums are the Catalan numbers (A000108). T(n,1)=2^n = A000079(n). T(n,n)=A089408(n+1). Sum(k*T(n,k),k=1..n)=A121686(n).


LINKS

Table of n, a(n) for n=1..55.


FORMULA

T(n,k)=2^(nk)*c(k/2)*binomial(n1,k1) if k is even and 2^(nk+1)*c((k1)/2)*binomial(n1,k1) if k is odd, where c(m)=binomial(2m,m)/(m+1) are the Catalan numbers (A000108). G.f.=(12z+2tz)(12zsqrt[(12z)^24t^2*z^2])/(2t^2*z^2)  1.


EXAMPLE

Triangle starts:
2;
4,1;
8,4,2;
16,12,12,2;
32,32,48,16,4;


MAPLE

c:=n>binomial(2*n, n)/(n+1): T:=proc(n, k) if k mod 2 = 0 then c(k/2)*binomial(n1, k1)*2^(nk) else c((k1)/2)*binomial(n1, k1)*2^(nk+1) fi end: for n from 1 to 11 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form


CROSSREFS

Cf. A000108, A000079, A089408, A121686.
Sequence in context: A182896 A207605 A112931 * A125810 A226504 A152195
Adjacent sequences: A121682 A121683 A121684 * A121686 A121687 A121688


KEYWORD

nonn,tabl


AUTHOR

Emeric Deutsch, Aug 15 2006


STATUS

approved



