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A116881 Row sums of triangle A116880 (generalized Catalan C(1,2)).

%I A116881 #26 Aug 05 2025 10:04:07
%S A116881 1,4,23,150,1037,7408,54035,399850,2990105,22540260,170991647,
%T A116881 1303789534,9983164453,76711854040,591236890667,4568611684306,
%U A116881 35382196437041,274564234870732,2134337640202295,16617270658727878
%N A116881 Row sums of triangle A116880 (generalized Catalan C(1,2)).
%H A116881 G. C. Greubel, <a href="/A116881/b116881.txt">Table of n, a(n) for n = 0..1000</a>
%F A116881 a(n) = Sum_{m=0..n} A116880(n,m), n>=0.
%F A116881 G.f.: (32*x^2+12*sqrt(1-8*x)*x-4*x)/(-32*x^3+sqrt(1-8*x)*(8*x^2+7*x-1)-36*x^2-3*x+1). - _Vladimir Kruchinin_, Nov 23 2014
%F A116881 a(n) = sum(k=0..n, ((k+1)^2*binomial(2*(n+1),n-k)*binomial(n+k+2,n+1))/((n+k+1)*(n+k+2))). - _Vladimir Kruchinin_, Nov 23 2014
%F A116881 a(n) ~ 2^(3*n+3) / (9*sqrt(Pi*n)). - _Vaclav Kotesovec_, Nov 23 2014
%F A116881 Conjecture: n*(3*n-4)*a(n) +(-21*n^2+43*n-10)*a(n-1) -4*(3*n-1)*(2*n-1)*a(n-2)=0. - _R. J. Mathar_, Jun 22 2016
%F A116881 From _Seiichi Manyama_, Aug 05 2025: (Start)
%F A116881 a(n) = Sum_{k=0..n} binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
%F A116881 a(n) = [x^n] (1+x)^(2*n+1)/(1-x)^n.
%F A116881 a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^n).
%F A116881 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(2*n+1,k).
%F A116881 a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(n+k-1,k). (End)
%t A116881 CoefficientList[Series[(32 x^2 + 12 Sqrt[1 - 8 x] x - 4 x) / (-32 x^3 + Sqrt[1 - 8 x] (8 x^2 + 7 x - 1) - 36 x^2 - 3 x + 1), {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 23 2014 *)
%o A116881 (Maxima) a(n):=sum(((k+1)^2*binomial(2*(n+1),n-k)*binomial(n+k+2,n+1))/((n+k+1)*(n+k+2)),k,0,n); /* _Vladimir Kruchinin_, Nov 23 2014 */
%Y A116881 Cf. A116880, A119259, A178792.
%K A116881 nonn,easy
%O A116881 0,2
%A A116881 _Wolfdieter Lang_, Mar 24 2006