|
|
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
|
|
|
LINKS
|
|
|
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
|
|
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
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
Jean Pallo (pallo(AT)u-bourgogne.fr)
|
|
STATUS
|
approved
|
|
|
|