A116881 Row sums of triangle A116880 (generalized Catalan C(1,2)).
1, 4, 23, 150, 1037, 7408, 54035, 399850, 2990105, 22540260, 170991647, 1303789534, 9983164453, 76711854040, 591236890667, 4568611684306, 35382196437041, 274564234870732, 2134337640202295, 16617270658727878
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
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 *) -
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 */
Formula
a(n) = Sum_{m=0..n} A116880(n,m), n>=0.
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
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
a(n) ~ 2^(3*n+3) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 23 2014
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
From Seiichi Manyama, Aug 05 2025: (Start)
a(n) = Sum_{k=0..n} binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
a(n) = [x^n] (1+x)^(2*n+1)/(1-x)^n.
a(n) = [x^n] 1/((1-x)^2 * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * (n-k+1) * binomial(2*n+1,k).
a(n) = Sum_{k=0..n} 2^k * (n-k+1) * binomial(n+k-1,k). (End)