A053137 Binomial coefficients C(2*n+8,8).

1, 45, 495, 3003, 12870, 43758, 125970, 319770, 735471, 1562275, 3108105, 5852925, 10518300, 18156204, 30260340, 48903492, 76904685, 118030185, 177232627, 260932815, 377348994, 536878650, 752538150, 1040465790, 1420494075, 1916797311, 2558620845, 3381098545

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn
• easy

Comments

Even-indexed members of ninth column of Pascal's triangle A007318.

Number of standard tableaux of shape (2n+1,1^8). - Emeric Deutsch, May 30 2004

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Milan Janjić, Two Enumerative Functions.
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A007318, A053136, A000581, A053130.

Programs

Magma

[Binomial(2*n+8,8): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011

Mathematica

Table[Binomial[2*n+8, 8], {n, 0, 30}] (* G. C. Greubel, Sep 03 2018 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,45,495,3003,12870,43758,125970,319770,735471},30] (* Harvey P. Dale, Jul 02 2022 *)

PARI

a(n)=binomial(2*n+8,8) \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = binomial(2*n+8, 8) = A000581(2*n+8).

G.f.: (1+36*x+126*x^2+84*x^3+9*x^4) / (1-x)^9 = (1+3*x) * (3*x^3+27*x^2+33*x+1) / (1-x)^9.

From Amiram Eldar, Nov 03 2022: (Start)

Sum_{n>=0} 1/a(n) = 512*log(2) - 5308/15.

Sum_{n>=0} (-1)^n/a(n) = 16*Pi + 32*log(2) - 1072/15. (End)