A002299 Binomial coefficients C(2*n+5,5).

1, 21, 126, 462, 1287, 3003, 6188, 11628, 20349, 33649, 53130, 80730, 118755, 169911, 237336, 324632, 435897, 575757, 749398, 962598, 1221759, 1533939, 1906884, 2349060, 2869685, 3478761, 4187106, 5006386, 5949147, 7028847, 8259888, 9657648, 11238513

Offset: 0

Author

N. J. A. Sloane, Eric Lane

Keywords

• nonn
• easy

Comments

Number of standard tableaux of shape (2n+1,1^5). - Emeric Deutsch, May 30 2004

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000 (terms 0..200 from Vincenzo Librandi)
• Milan Janjic, Two Enumerative Functions.
• J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math., Vol. 201, No. 1 (2006), pp. 143-179. [Th. 7.2(i), case a=1]
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000389, A053127.

Programs

Magma

[Binomial(2*n+5,5): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011

Mathematica

Table[Binomial[2*n + 5, 5], {n, 0, 50}] (* G. C. Greubel, Nov 23 2017 *)

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

Formula

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

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

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

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

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

From Peter Bala, Sep 03 2023: (Start)

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

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

a(n) = 2*A225007(n) - A006324(n+1). - Yasser Arath Chavez Reyes, Feb 27 2024