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%I #45 Nov 22 2024 08:53:56
%S 1,11,78,455,2380,11628,54264,245157,1081575,4686825,20030010,
%T 84672315,354817320,1476337800,6107086800,25140840660,103077446706,
%U 421171648758,1715884494940,6973199770790,28277527346376,114456658306760,462525733568080,1866442158555975
%N a(n) = binomial(2n+1,n-4).
%H Robert Israel, <a href="/A030054/b030054.txt">Table of n, a(n) for n = 4..1661</a>
%H Milan Janjic, <a href="https://web.archive.org/web/20181001110015/https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%F G.f.: x^4*512/((1-sqrt(1-4*x))^9*sqrt(1-4*x))+(-1/x^5+7/x^4-15/x^3+10/x^2-1/x). - _Vladimir Kruchinin_, Aug 11 2015
%F From _Robert Israel_, Jun 11 2019: (Start)
%F (54 + 36*n)*a(n) + (-438 - 129*n)*a(n + 1) + (714 + 138*n)*a(n + 2) + (-432 - 63*n)*a(n + 3) + (110 + 13*n)*a(n + 4) + (-10 - n)*a(n + 5) = 0.
%F a(n) ~ 2^(2*n+1)/sqrt(n*Pi). (End)
%F From _Amiram Eldar_, Jan 24 2022: (Start)
%F Sum_{n>=4} 1/a(n) = 317/210 - 2*Pi/(9*sqrt(3)).
%F Sum_{n>=4} (-1)^n/a(n) = 2908*log(phi)/(5*sqrt(5)) - 8697/70, where phi is the golden ratio (A001622). (End)
%F G.f.: 2F1([11/2,5],[10],4*x). - _Karol A. Penson_, Apr 24 2024
%F From _Peter Bala_, Oct 13 2024: (Start)
%F a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * sqrt(x)*(x^4 - 9*x^3 + 27*x^2 - 30*x + 9)/sqrt((4 - x)).
%F G.f. x^4 * B(x) * C(x)^9, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
%F D-finite with recurrence -(n+5)*(n-4)*a(n) +2*n*(2*n+1)*a(n-1)=0. - _R. J. Mathar_, Nov 22 2024
%p seq(binomial(2*n+1,n-4),n=4..50); # _Robert Israel_, Jun 11 2019
%t Table[Binomial[2n+1,n-4],{n,4,40}] (* _Harvey P. Dale_, Mar 31 2011 *)
%o (PARI) vector(30, n, m=n+4; binomial(2*m+1,m-4)) \\ _Michel Marcus_, Aug 11 2015
%Y Diagonal 10 of triangle A100257.
%Y Fifth unsigned column (s=4) of A113187. - _Wolfdieter Lang_, Oct 19 2012
%Y Cf. A001622.
%Y Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A003516 (m = 5), A002696 (m = 6), A030053 - A030056, A004310 - A004318.
%K nonn
%O 4,2
%A _N. J. A. Sloane_