A011274 Triangle of numbers of hybrid rooted trees (divided by Fibonacci numbers).

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

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: A317360 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