A179381 Row sums of A179318.

1, 2, 4, 10, 26, 78, 236, 770, 2520, 8606, 29364, 103302, 362226, 1298882, 4645670, 16897224, 61296686, 225457006, 826950080, 3067763394, 11353597198, 42414220022, 158095481910, 594108418428, 2227714454332, 8412269224862, 31704876569698, 120223392641084, 455053649594196, 1731861709709542, 6579658381972974

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

Cf. A000108, A318264.

Sequence in context: A148101 A052854 A148102 * A096807 A003239 A195924

Adjacent sequences: A179378 A179379 A179380 * A179382 A179383 A179384

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