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A114506
Triangle read by rows: T(n,k) is number of Dyck paths of semilength n having k ascents of length 3 (0<=k<=floor(n/3)). Also number of ordered trees with n edges that have k vertices of outdegree 3.
2
1, 1, 2, 4, 1, 10, 4, 27, 15, 79, 50, 3, 240, 168, 21, 750, 568, 112, 2387, 1959, 504, 12, 7711, 6850, 2115, 120, 25214, 24211, 8536, 825, 83315, 86164, 33858, 4620, 55, 277799, 308152, 133068, 23166, 715, 933596, 1106028, 520338, 108472, 6006
OFFSET
0,3
COMMENTS
Row n has 1+floor(n/3) terms. Row sums yield the Catalan numbers (A000108). Column 0 yields A114507. Sum(kT(n,k),k=0..floor(n/3))=binomial(2n-4,n-3) (A001791).
FORMULA
G.f. G=G(t, z) satisfies (1-t)z^4*G^4-(1-t)z^3*G^3+zG^2-G+1=0.
EXAMPLE
T(4,1)=4 because we have UDUUUDDD, UUUDDDUD, UUUDUDDD and UUUDDUDD, where U=(1,1), D=(1,-1).
Triangle starts:
1;
1;
2;
4,1;
10,4;
27,15;
79,50,3;
240,168,21;
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 03 2005
STATUS
approved