A271823 a(n) = binomial(2*n-4,n-1)*(n+3)/n.
4, 0, 2, 7, 24, 84, 300, 1089, 4004, 14872, 55692, 209950, 795872, 3031032, 11589240, 44462565, 171085500, 660009840, 2551955340, 9887121090, 38374857840, 149184555000, 580807904040, 2264193450090, 8837215647624, 34529741789904, 135054066707000
Offset: 1
Keywords
Programs
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Magma
[Binomial(2*n-4,n-1)*(n+3)/n: n in [1..30]]; // Vincenzo Librandi, Apr 15 2016
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Mathematica
Table[Binomial[2 n - 4, n - 1] (n + 3) / n, {n, 45}] (* Vincenzo Librandi, Apr 15 2016 *) -
Maxima
taylor((-6*x^2+sqrt(1-4*x)*(4*x+1)+6*x-1)/(2*sqrt(1-4*x)),x,0,27);
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PARI
lista(nn) = for(n=1, nn, print1(binomial(2*n-4, n-1)*(n+3)/n, ", ")); \\ Altug Alkan, Apr 15 2016
Formula
G.f.: (-6*x^2+sqrt(1-4*x)*(4*x+1)+6*x-1)/(2*sqrt(1-4*x)).
D-finite with recurrence: n*(n+2)*(n-3)*a(n) -2*(n-2)*(2*n-5)*(n+3)*a(n-1)=0. - R. J. Mathar, Jun 07 2016
D-finite with recurrence: n*a(n) +2*(-5*n+7)*a(n-1) +6*(5*n-14)*a(n-2) +12*(-2*n+9)*a(n-3)=0. - R. J. Mathar, Jan 27 2020
From Amiram Eldar, Feb 17 2023: (Start)
Sum_{n>=3} 1/a(n) = 168*Pi^2 - 82262*Pi/(45*sqrt(3)) + 248747/150.
Sum_{n>=3} (-1)^(n+1)/a(n) = 65084*log(phi)/(5*sqrt(5)) - 6048*log(phi)^2 - 70019/50, where phi is the golden ratio (A001622). (End)
a(n) ~ 4^(n-2)/sqrt(n*Pi). - Stefano Spezia, Jan 30 2025