OFFSET
0,3
COMMENTS
The analogous sequence with associahedron in place of permutohedron is (n+1)^{n-1}.
This also counts maximal chains in the shard intersection orders of type A, see Theorem 1.3 in Reading reference. - F. Chapoton, Apr 29 2015
The ordinary generating series may be Gevrey of order 2, i.e., the coefficients may be bounded by A*B^n*(n!)^2 for some constants A and B. - F. Chapoton, Jul 07 2023
LINKS
M. F. Hasler, Table of n, a(n) for n = 0..100
Jean-Louis Loday, More information
Jean-Louis Loday, Parking functions and triangulation of the associahedron, arXiv:math/0510380 [math.AT], 2005.
N. Reading, Noncrossing partitions and the shard intersection order, arXiv:0909.3288 [math.CO], 2009.
N. Reading, Noncrossing partitions and the shard intersection order, J. Algebraic Combin. 33 (2011), no. 4.
FORMULA
a(n) = Sum_{m=0..n-1} (binomial(n+1, m+1) - 1)*binomial(n-1, m)*a(m)*a(n-m-1). - Robert G. Wilson v, Oct 31 2005
MAPLE
function y=binom(n, p); y=1; for j = 0 : p-1; y=y*(n-j); end; for j = 1 : p; y=y/j; end; format long; nmax = 14; mm=nmax+1; zp=zeros(mm, 1); zp(1:1) = 1; for n = 1 : nmax; z=0; for p = 0 : n-1; z=z+ (binom(n+1, p+1)-1) * binom(n-1, p) * zp(p+1:p+1) * zp(n-p:n-p); end; zp(n+1:n+1)=z; z; end; n, z
MATHEMATICA
f[0] = 1; f[n_] := Sum[(Binomial[n + 1, m + 1] - 1)Binomial[n - 1, m]f[m]f[n - m - 1], {m, 0, n - 1}]; Table[f[n], {n, 0, 16}] (* Robert G. Wilson v, Oct 31 2005 *)
PROG
(PARI) a111169=[1]; A111169(n)={n<3&&return(n^n); global(a111169); while(n>m=#a111169, a111169=concat(a111169, sum(k=1, m-1, (binomial(m+2, k+1)-1)*binomial(m, k)*a111169[k]*a111169[m-k], 2*(m+1)*a111169[m]))); a111169[n]} \\ M. F. Hasler, May 02 2015
(Sage)
@cached_function
def a(n):
if n == 0:
return 1
return sum((binomial(n + 1, m + 1) - 1) * binomial(n - 1, m)
* a(m) * a(n - m - 1) for m in range(n))
# F. Chapoton, Jul 07 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jean-Louis Loday (loday(AT)math.u-strasbg.fr), Oct 21 2005
EXTENSIONS
More terms from Robert G. Wilson v, Oct 31 2005
STATUS
approved