%I #26 Sep 08 2022 08:44:50

%S 1,13,105,680,3876,20349,100947,480700,2220075,10015005,44352165,

%T 193536720,834451800,3562467300,15084504396,63432274896,265182149218,

%U 1103068603890,4568648125690,18851684897584

%N a(n) = binomial(2*n+1, n-5).

%H G. C. Greubel, <a href="/A030055/b030055.txt">Table of n, a(n) for n = 5..1000</a>

%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.

%F G.f.: x^5*2048/((1-sqrt(1-4*x))^11*sqrt(1-4*x))+(-1/x^6+9/x^5-28/x^4+35/x^3-15/x^2+1/x). - _Vladimir Kruchinin_, Aug 11 2015

%F From _Amiram Eldar_, Jan 24 2022: (Start)

%F Sum_{n>=5} 1/a(n) = 9497/1260 - 32*Pi/(9*sqrt(3)).

%F Sum_{n>=5} (-1)^(n+1)/a(n) = 9392*log(phi)/(5*sqrt(5)) - 508169/1260, where phi is the golden ratio (A001622). (End)

%p seq(binomial(2*n+1,n-5),n=5..25); # _Muniru A Asiru_, Oct 24 2018

%t Table[Binomial[2*n+1, n-5], {n, 5, 30}] (* _G. C. Greubel_, Oct 23 2018 *)

%o (PARI) vector(30, n, m=n+4; binomial(2*m+1,m-5)) \\ _Michel Marcus_, Aug 11 2015

%o (Magma) [Binomial(2*n+1, n-5): n in [5..30]]; // _G. C. Greubel_, Oct 23 2018

%o (GAP) List([5..25],n->Binomial(2*n+1,n-5)); # _Muniru A Asiru_, Oct 24 2018

%Y Diagonal 12 of triangle A100257.

%Y Cf. A001622.

%K nonn

%O 5,2

%A _N. J. A. Sloane_