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A179381 Row sums of A179318.

%I A179381 #40 Apr 09 2025 11:37:21
%S A179381 1,2,4,10,26,78,236,770,2520,8606,29364,103302,362226,1298882,4645670,
%T A179381 16897224,61296686,225457006,826950080,3067763394,11353597198,
%U A179381 42414220022,158095481910,594108418428,2227714454332,8412269224862,31704876569698,120223392641084,455053649594196,1731861709709542,6579658381972974
%N A179381 Row sums of A179318.
%H A179381 Vaclav Kotesovec, <a href="/A179381/b179381.txt">Table of n, a(n) for n = 1..1650</a>
%H A179381 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 A179381 G.f.: -1 + Product_{n>=1} 1/(1-C(n-1)*x^n), where C(n) = A000108(n). - _Vladimir Kruchinin_, Aug 18 2014
%F A179381 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
%F A179381 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 A179381 The table has shape A000041 and begins:
%e A179381   1
%e A179381   1 1
%e A179381   2 1 1
%e A179381   5 2 1 1 1
%e A179381   14 5 2 2 1 1 1
%e A179381 so a(n) begins 1 2 4 10 26 ...
%o A179381 (PARI)
%o A179381 N = 66;  x = 'x +O('x^N);
%o A179381 C(n) = binomial(2*n,n)/(n+1);
%o A179381 gf = -1 + 1/prod(n=1, N, 1 - C(n-1)*x^n );
%o A179381 Vec(gf)
%o A179381 \\ _Joerg Arndt_, Aug 18 2014
%o A179381 (Maxima)
%o A179381 C(n):=  1/(n+1)*binomial(2*n,n);
%o A179381 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 A179381 makelist(s(1,n),n,1,27);  /* _Vladimir Kruchinin_, Sep 06 2014 */
%Y A179381 Cf. A000108, A318264.
%K A179381 easy,nonn
%O A179381 1,2
%A A179381 _Alford Arnold_, Jul 12 2010
%E A179381 Terms 8606 and beyond (using Kruchinin's formula) by _Joerg Arndt_, Aug 18 2014