OFFSET
1,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..1650
StackExchange, Infinite product with the Catalan numbers, Mar 12 2018
FORMULA
G.f.: -1 + prod(n>=1, 1/(1-C(n-1)*x^n), where C(n) = A000108(n). - Vladimir Kruchinin, Aug 18 2014
a(n) = s(1,n), where s(m,n) = C(n-1)+sum(k=m..n/2, C(k-1)*s(k,n-k), a(n,n) = C(n-1), C(n) are the Catalan numbers (A000108). - Vladimir Kruchinin, Sep 06 2014
a(n) ~ c * 4^n / n^(3/2), where c = 1 / (4*sqrt(Pi) * Product_{k>=1} (1 - binomial(2*k-2,k-1) / (k * 4^k))) = 0.2422046382280667... - Vaclav Kotesovec, Mar 08 2018
EXAMPLE
The table has shape A000041 and begins:
1
1 1
2 1 1
5 2 1 1 1
14 5 2 2 1 1 1
so
a(n) begins 1 2 4 10 26 ...
PROG
(PARI)
N = 66; x = 'x +O('x^N);
C(n) = binomial(2*n, n)/(n+1);
gf = -1 + 1/prod(n=1, N, 1 - C(n-1)*x^n );
Vec(gf)
\\ Joerg Arndt, Aug 18 2014
(Maxima)
C(n):= 1/(n+1)*binomial(2*n, n);
s(m, n):=if m>n then 0 else if n=m then C(n-1) else sum(C(k-1)*s(k, n-k), k, m, ceiling(n/2))+C(n-1);
makelist(s(1, n), n, 1, 27); /* Vladimir Kruchinin, Sep 06 2014 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Alford Arnold, Jul 12 2010
EXTENSIONS
Terms 8606 and beyond (using Kruchinin's formula) by Joerg Arndt, Aug 18 2014
STATUS
approved