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%I #31 Aug 23 2018 08:13:44
%S 1,2,4,10,26,78,236,770,2520,8606,29364,103302,362226,1298882,4645670,
%T 16897224,61296686,225457006,826950080,3067763394,11353597198,
%U 42414220022,158095481910,594108418428,2227714454332,8412269224862,31704876569698,120223392641084,455053649594196,1731861709709542,6579658381972974
%N Row sums of A179318.
%H Vaclav Kotesovec, <a href="/A179381/b179381.txt">Table of n, a(n) for n = 1..1650</a>
%H StackExchange, <a href="https://mathematica.stackexchange.com/questions/167590/infinite-product-with-the-catalan-numbers">Infinite product with the Catalan numbers</a>, Mar 12 2018
%F G.f.: -1 + prod(n>=1, 1/(1-C(n-1)*x^n), where C(n) = A000108(n). - _Vladimir Kruchinin_, Aug 18 2014
%F 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
%F 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
%e The table has shape A000041 and begins:
%e 1
%e 1 1
%e 2 1 1
%e 5 2 1 1 1
%e 14 5 2 2 1 1 1
%e so
%e a(n) begins 1 2 4 10 26 ...
%o (PARI)
%o N = 66; x = 'x +O('x^N);
%o C(n) = binomial(2*n,n)/(n+1);
%o gf = -1 + 1/prod(n=1, N, 1 - C(n-1)*x^n );
%o Vec(gf)
%o \\ _Joerg Arndt_, Aug 18 2014
%o (Maxima)
%o C(n):= 1/(n+1)*binomial(2*n,n);
%o 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);
%o makelist(s(1,n),n,1,27); /* _Vladimir Kruchinin_, Sep 06 2014 */
%Y Cf. A000108, A318264.
%K easy,nonn
%O 1,2
%A _Alford Arnold_, Jul 12 2010
%E Terms 8606 and beyond (using Kruchinin's formula) by _Joerg Arndt_, Aug 18 2014
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