|
| |
|
|
A153391
|
|
G.f.: A(x) = F(x*G(x)^2) where F(x) = G(x*F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan).
|
|
2
|
|
|
|
1, 1, 5, 29, 183, 1223, 8525, 61366, 453003, 3412077, 26124599, 202748728, 1591450129, 12612760009, 100790253764, 811227147197, 6570431009209, 53512143110041, 437976298197769, 3600504527707557, 29716593448484673
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(2n,n-k)*k/n for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^2*A(x)^3 where G(x) is the g.f. of A000108.
G.f. satisfies: A(x*F(x)) = F(F(x)-1) where F(x) is the g.f. of A001764.
|
|
|
EXAMPLE
|
G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 29*x^3 + 183*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 +...
A(x)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 449*x^4 + 3102*x^5 +...
A(x)^3 = 1 + 3*x + 18*x^2 + 118*x^3 + 813*x^4 + 5799*x^5 +...
G(x)^2*A(x)^3 = 1 + 5*x + 29*x^2 + 183*x^3 + 1223*x^4 + 8525*x^5 +...
|
|
|
PROG
|
(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*binomial(2*(n-k)+2*k, n-k)*2*k/(2*(n-k)+2*k)))}
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
