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

Author

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

Keywords

• nonn
• easy
• tabl
• nice

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

Examples

			     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
		

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

Crossrefs

Cf. A000045, A011270, A011272.

Programs

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 *)

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 */

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