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A072247 Triangle T(n,k) (n >= 2, 2 <= k <= n-1 if n > 2) giving number of non-crossing trees with n nodes and k endpoints. 3
1, 3, 8, 4, 20, 30, 5, 48, 144, 75, 6, 112, 560, 595, 154, 7, 256, 1920, 3440, 1848, 280, 8, 576, 6048, 16380, 14994, 4788, 468, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

For n > 2 the n-th row has n-2 terms.

The difference between this sequence and A091320 is that this sequence considers the degrees of all vertices whereas A091320 ignores the degree of the root vertex. - Andrew Howroyd, Nov 06 2017

LINKS

Table of n, a(n) for n=2..30.

E. Deutsch and M. Noy, Statistics on non-crossing trees, Discrete Math., 254 (2002), 75-87.

FORMULA

T(n, k) = U(n, k-1) - U(n, k) + binomial(n-1, k)*Sum_{j=0..k-1} binomial(n-1, j)*binomial(n-k-1, k-1-j)*2^(n-2*k+j)/(n-1) where U(n,k) = 2*binomial(n-2, k)*Sum_{j=0..k-1} binomial(n-1, j)*binomial(n-k-2, k-1-j)*2^(n-1-2*k+j)/(n-2) for 2 < k < n. - Andrew Howroyd, Nov 06 2017

EXAMPLE

Triangle begins:

   1;

   3;

   8,   4;

  20,  30,  5;

  48, 144, 75, 6;

  ...

PROG

(PARI)

U(n, k) = 2*binomial(n-2, k)*sum(j=0, k-1, binomial(n-1, j)*binomial(n-k-2, k-1-j)*2^(n-1-2*k+j))/(n-2);

W(n, k) = binomial(n-1, k)*sum(j=0, k-1, binomial(n-1, j)*binomial(n-k-1, k-1-j)*2^(n-2*k+j))/(n-1);

T(n, k) = if(n<3, n==2&&k==2, if(1<k&&k<n, U(n, k-1)-U(n, k)+W(n, k)));

for(n=2, 10, for(k=2, n-(n>2), print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 06 2017

CROSSREFS

Column k=2 gives A001792, row sums are A001764.

Cf. A091320.

Sequence in context: A212886 A127438 A106292 * A051359 A016670 A021726

Adjacent sequences:  A072244 A072245 A072246 * A072248 A072249 A072250

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Jul 06 2002

EXTENSIONS

Offset corrected by Andrew Howroyd, Nov 06 2017

STATUS

approved

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Last modified February 24 12:59 EST 2018. Contains 299623 sequences. (Running on oeis4.)