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A127157 Triangle read by rows: T(n,k) is the number of ordered trees with n edges and 2k nodes of odd degree (not outdegree; 1<=k<=ceil(n/2)). 0
1, 2, 3, 2, 4, 10, 5, 30, 7, 6, 70, 56, 7, 140, 252, 30, 8, 252, 840, 330, 9, 420, 2310, 1980, 143, 10, 660, 5544, 8580, 2002, 11, 990, 12012, 30030, 15015, 728, 12, 1430, 24024, 90090, 80080, 12376, 13, 2002, 45045, 240240, 340340, 111384, 3876, 14, 2730 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row n has ceil(n/2) terms. Row sums are the Catalan numbers (A000108). T(n,1)=n; T(n,2)=2*binom(n+1,4)=2*A000332(n+1); T(n,3)=7*binom(n+2,7)=7*A000580(n+2); T(n,4)=30*binom(n+3,10)=30*A001287(n+3); T(n,5)=143*binom(n+4,13)=143*A010966(n+4); T(2n-1,n)=A006013(n-1).

T(n,k) is the number of ordered trees (A000108) with n edges, exactly k of whose vertices possess at least one leaf child. [David Callan, Aug 22 2014]

LINKS

Table of n, a(n) for n=1..51.

J.-C. Aval, A. Boussicault, B. Delcroix-Oger, F. Hivert, et al., Non-ambiguous trees: new results and generalization, arXiv preprint arXiv:1511.09455 [math.CO], 2015.

FORMULA

T(n,k)=2*binomial(3k-1,2k)*binomial(n-1+k,3k-2)/(3k-1) (formula obtained only by inspection). G.f.=G-1, where G=G(t,z) satisfies z^2*G^3-z(z+2)G^2+(1+2z)G-t^2*z-1=0.

EXAMPLE

Triangle starts:

1;

2;

3,2;

4,10;

5,30,7;

6,70,56;

MAPLE

T:=(n, k)->2*binomial(3*k-1, 2*k)*binomial(n-1+k, 3*k-2)/(3*k-1): for n from 1 to 15 do seq(T(n, k), k=1..ceil(n/2)) od;

MATHEMATICA

m = 14(*rows*); G = 0; Do[G = Series[(1 + t^2 z - G^3 z^2 + G^2 z (2+z))/ (1+2z), {t, 0, m}, {z, 0, m}] // Normal // Expand, m]; Rest[ CoefficientList[#, t^2]]& /@ Rest[CoefficientList[G-1, z] ] // Flatten (* Jean-Fran├žois Alcover, Jan 23 2019 *)

CROSSREFS

Cf. A000108, A000332, A000580, A001287, A010966, A006013.

Sequence in context: A304489 A034800 A082771 * A236406 A247497 A202714

Adjacent sequences:  A127154 A127155 A127156 * A127158 A127159 A127160

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Feb 27 2007

STATUS

approved

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Last modified May 21 08:53 EDT 2019. Contains 323441 sequences. (Running on oeis4.)