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
Examples
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 ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 1..1650
- StackExchange, Infinite product with the Catalan numbers, Mar 12 2018.
Programs
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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 */
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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
Formula
G.f.: -1 + Product_{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), s(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
Extensions
Terms 8606 and beyond (using Kruchinin's formula) by Joerg Arndt, Aug 18 2014