|
|
A052723
|
|
Expansion of e.g.f. (1 - x - sqrt(1-2*x+x^2-4*x^3))/(2*x).
|
|
9
|
|
|
0, 0, 2, 6, 24, 240, 2880, 35280, 524160, 9434880, 188697600, 4151347200, 101548339200, 2727435110400, 79332244992000, 2488504322304000, 83879464660992000, 3021209014247424000, 115754916599562240000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
FORMULA
|
D-finite with recurrence: a(0) = a(1) = 0, a(2) = 2, a(3) = 6, a(4) = 24, (n+4)*a(n+3) = (15 + 11*n + 2*n^2)*a(n+2) - (6 + 11*n + 6*n^2 + n^3)*a(n+1) - (12 - 2*n - 32*n^2 - 22*n^2 - 4*n^4)*a(n).
|
|
MAPLE
|
spec := [S, {B=Prod(S, S), C=Union(B, S, Z), S=Prod(Z, C)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(n!*add(binomial(n-2-k, 2*k)*binomial(2*k, k)/(k+1), k=0..floor((n-2)/3)), n=0..18); # Mark van Hoeij, May 12 2013
|
|
MATHEMATICA
|
With[{nn=20}, CoefficientList[Series[(1-x-Sqrt[1-2x+x^2-4x^3])/(2x), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jan 19 2017 *)
a[n_]:= a[n]= n!*Sum[Binomial[n-k-2, 2*k]*CatalanNumber[k], {k, 0, Floor[(n-2)/2]}];
|
|
PROG
|
(SageMath)
def A052723(n): return factorial(n)*sum( binomial(n-k-2, 2*k)*catalan_number(k) for k in (0..(n-2)//2) )
|
|
CROSSREFS
|
Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722.
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
|
|
STATUS
|
approved
|
|
|
|