|
| |
|
|
A254314
|
|
Hankel transform of a(n) is A006720(n). Hankel transform of a(n+1) is A006720(n+2).
|
|
1
|
|
|
|
1, 1, 2, 5, 14, 43, 143, 507, 1887, 7279, 28828, 116455, 477709, 1983779, 8321474, 35203777, 150014157, 643302743, 2773997104, 12020733635, 52319374842, 228616865437, 1002544803949, 4410700121313, 19462407890220, 86111960348939, 381956399941011
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
a(n+1) is the number of rooted plane trees with integer compositions labeling the leaves (empty labels are allowed), with total size n. The total size is the number of edges in the tree plus the sum of the sizes of the integer compositions labeling the leaves.
Example: a(3)=5 because there are 5 elements of size 2: two trees that consist of the root and no descendants, hence the root is itself a leaf and it can be labeled by either 2=2 or by 1=1+1, then a tree with the root and one descendant that is a leaf labeled with 1=1, then a tree with the root and two descendants with no labels on the leaves, and finally a tree with the root with one descendant with a descendant that is a leaf with no label. - Ricardo Gómez Aíza, Feb 29 2024
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f. A(x) satisfies 0 = (2*x-1)*A(x)^2 + (x^2-6*x+3)*A(x) + (3*x-2).
G.f.: (3 - 6*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 )) / (2*(1 - 2*x)).
Conjecture: n*a(n) +2*(-5*n+6)*a(n-1) +2*(17*n-39)*a(n-2) +6*(-8*n+27)*a(n-3) +(25*n-114)*a(n-4) +2*(-n+6)*a(n-5)=0. - R. J. Mathar, Jun 07 2016
a(n) ~ sqrt(b*(5-32*b+46*b^2))/(2*sqrt((1-2*b)^3*Pi*n^3))*(1/b)^n where b = (11-c-100/c)/3 and c = (-998+6*sqrt(111)*i)^(1/3). - Ricardo Gómez Aíza, Feb 29 2024
|
|
|
EXAMPLE
|
G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 43*x^5 + 143*x^6 + 507*x^7 + ...
|
|
|
MATHEMATICA
|
CoefficientList[Series[(3-6*x+x^2 - Sqrt[(1-4*x+x^2)^2 -4*x^3])/(2*(1 - 2*x)), {x, 0, 60}], x] (* G. C. Greubel, Aug 10 2018 *)
|
|
|
PROG
|
(PARI) {a(n) = if( n<0, 0, polcoeff( (3 - 6*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 + x^2 * O(x^n))) / (2*(1 - 2*x)), n))};
(Magma) m:=60; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3-6*x+x^2 - Sqrt((1-4*x+x^2)^2 -4*x^3))/(2*(1 - 2*x)))); // G. C. Greubel, Aug 10 2018
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
