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A011274 Triangle of numbers of hybrid rooted trees (divided by Fibonacci numbers). 1
1, 2, 1, 7, 4, 1, 31, 18, 6, 1, 154, 90, 33, 8, 1, 820, 481, 185, 52, 10, 1, 4575, 2690, 1065, 324, 75, 12, 1, 26398, 15547, 6276, 2006, 515, 102, 14, 1, 156233, 92124, 37711, 12468, 3420, 766, 133, 16, 1, 943174, 556664, 230277, 78030, 22412, 5439, 1085, 168, 18, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Triangle T(n,k) = [x^(n-k)] A(x)^k where A(x) is the o.g.f. of A007863. - Vladimir Kruchinin, Mar 17 2011

Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A007863. - Philippe Deléham, Feb 03 2014

LINKS

Vincenzo Librandi, Rows n = 1..100, flattened

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

J. M. Pallo, On the listing and random generation of hybrid binary trees, International Journal of Computer Mathematics, 50 (1994) 135-145.

Index entries for sequences related to rooted trees

FORMULA

T(n,k) = k/n *sum_{i=0..n-k} binomial(i+n-1,n-1)*binomial(i+n,n-k-i). - Vladimir Kruchinin, Mar 17 2011

r/(m*n+r) *T((m+1)*n+r,m*n+r) =sum_{k=1..n} k*T((m+1)*n-k,m*n) *T(r+k,r)/n. - Vladimir Kruchinin, Mar 17 2011

T(n,m) = (m/n) *sum_{k=1..n-m+1} k *A007863(k-1) *T(n-k,m-1), 1<m<=n. - Vladimir Kruchinin, Mar 17 2011

EXAMPLE

     1

     2    1

     7    4    1

    31   18    6   1

   154   90   33   8  1

   820  481  185  52 10  1

  4575 2690 1065 324 75 12 1

Production matrix is:

   2   1

   3   2   1

   5   3   2   1

   8   5   3   2   1

  13   8   5   3   2   1

  21  13   8   5   3   2   1

  34  21  13   8   5   3   2   1

  55  34  21  13   8   5   3   2   1

  89  55  34  21  13   8   5   3   2   1

  ... - Philippe Deléham, Feb 03 2014

MAPLE

A011274 := proc(n, k) k/n*add( binomial(i+n-1, n-1)*binomial(i+n, n-k-i), i=0..n-k) ; end proc: # R. J. Mathar, Mar 21 2011

MATHEMATICA

t[n_, k_] := k/n*Binomial[n, k]*HypergeometricPFQ[ {k-n, n, n+1}, {1/2 + k/2, 1+k/2}, -1/4]; Flatten[ Table[ t[n, k], {n, 1, 10}, {k, 1, n}]] (* Jean-François Alcover, Dec 02 2011, after Vladimir Kruchinin *)

PROG

(Maxima) A011274(n, k):= k/n*sum(binomial(i+n-1, n-1)*binomial(i+n, n-k-i), i, 0, n-k); /* Vladimir Kruchinin, Mar 17 2011 */

CROSSREFS

Cf. A000045, A011270, A011272.

Sequence in context: A072248 A177011 A092276 * A122843 A167196 A241881

Adjacent sequences:  A011271 A011272 A011273 * A011275 A011276 A011277

KEYWORD

nonn,easy,tabl,nice

AUTHOR

Jean Pallo (pallo(AT)u-bourgogne.fr)

STATUS

approved

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Last modified May 23 18:24 EDT 2017. Contains 286926 sequences.