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

%I A011274 #32 Oct 07 2017 02:35:20
%S A011274 1,2,1,7,4,1,31,18,6,1,154,90,33,8,1,820,481,185,52,10,1,4575,2690,
%T A011274 1065,324,75,12,1,26398,15547,6276,2006,515,102,14,1,156233,92124,
%U A011274 37711,12468,3420,766,133,16,1,943174,556664,230277,78030,22412,5439,1085,168,18,1
%N A011274 Triangle of numbers of hybrid rooted trees (divided by Fibonacci numbers).
%C A011274 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
%C A011274 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A007863. - _Philippe Deléham_, Feb 03 2014
%H A011274 Vincenzo Librandi, <a href="/A011274/b011274.txt">Rows n = 1..100, flattened</a>
%H A011274 Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%H A011274 J. M. Pallo, <a href="http://dx.doi.org/10.1080/00207169408804251">On the listing and random generation of hybrid binary trees</a>, International Journal of Computer Mathematics, 50 (1994) 135-145.
%H A011274 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A011274 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
%F A011274 (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
%F A011274 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
%e A011274      1
%e A011274      2    1
%e A011274      7    4    1
%e A011274     31   18    6   1
%e A011274    154   90   33   8  1
%e A011274    820  481  185  52 10  1
%e A011274   4575 2690 1065 324 75 12 1
%e A011274 Production matrix is:
%e A011274    2   1
%e A011274    3   2   1
%e A011274    5   3   2   1
%e A011274    8   5   3   2   1
%e A011274   13   8   5   3   2   1
%e A011274   21  13   8   5   3   2   1
%e A011274   34  21  13   8   5   3   2   1
%e A011274   55  34  21  13   8   5   3   2   1
%e A011274   89  55  34  21  13   8   5   3   2   1
%e A011274   ... - _Philippe Deléham_, Feb 03 2014
%p A011274 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
%t A011274 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_ *)
%o A011274 (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 */
%Y A011274 Cf. A000045, A011270, A011272.
%K A011274 nonn,easy,tabl,nice
%O A011274 1,2
%A A011274 Jean Pallo (pallo(AT)u-bourgogne.fr)