A002299 - OEIS

%I #61 Mar 22 2024 17:48:00

%S 1,21,126,462,1287,3003,6188,11628,20349,33649,53130,80730,118755,

%T 169911,237336,324632,435897,575757,749398,962598,1221759,1533939,

%U 1906884,2349060,2869685,3478761,4187106,5006386,5949147,7028847,8259888,9657648,11238513

%N Binomial coefficients C(2*n+5,5).

%C Number of standard tableaux of shape (2n+1,1^5). - _Emeric Deutsch_, May 30 2004

%H G. C. Greubel, <a href="/A002299/b002299.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..200 from Vincenzo Librandi)

%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.

%H J. M. Landsberg and L. Manivel, <a href="https://doi.org/10.1016/j.aim.2005.02.001">The sextonions and E7 1/2</a>, Adv. Math., Vol. 201, No. 1 (2006), pp. 143-179. [Th. 7.2(i), case a=1]

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F a(n) = A000389(2*n+5).

%F G.f.: (1+15*x+15*x^2+x^3)/(1-x)^6 = (1+x)*(x^2+14*x+1)/(1-x)^6.

%F E.g.f.: (30 + 600*x + 1275*x^2 + 730*x^3 + 140*x^4 + 8*x^5)*exp(x)/30. - _G. C. Greubel_, Nov 23 2017

%F Sum_{n>=0} (-1)^n/a(n) = 5*(10/3 - Pi). - _Matthieu Pluntz_, Oct 08 2019

%F Sum_{n>=0} 1/a(n) = 40*log(2) - 80/3. - _Amiram Eldar_, Jan 03 2022

%F From _Peter Bala_, Sep 03 2023: (Start)

%F a(n) = Sum_{0 <= i <= j <= n} (j+1)*(2*i+1)^2.

%F a(n) = (n+2)*(2*n+5)/(n*(2*n-1))*a(n-1) with a(0) := 1. (End)

%F a(n) = 2*A225007(n) - A006324(n+1). - _Yasser Arath Chavez Reyes_, Feb 27 2024

%t Table[Binomial[2*n + 5, 5], {n, 0, 50}] (* _G. C. Greubel_, Nov 23 2017 *)

%o (Magma) [Binomial(2*n+5,5): n in [0..30]]; // _Vincenzo Librandi_, Oct 07 2011

%o (PARI) a(n)=n*(8*n^4+60*n^3+170*n^2+225*n+137)/30+1 \\ _Charles R Greathouse IV_, Apr 18 2012

%Y Cf. A000389, A053127.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Eric Lane