A053129 Binomial coefficients C(2*n-6,7).

8, 120, 792, 3432, 11440, 31824, 77520, 170544, 346104, 657800, 1184040, 2035800, 3365856, 5379616, 8347680, 12620256, 18643560, 26978328, 38320568, 53524680, 73629072, 99884400, 133784560, 177100560, 231917400, 300674088, 386206920, 491796152, 621216192

Offset: 7

Author

Wolfdieter Lang

Keywords

• nonn
• easy

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 7..200
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Milan Janjić, Two Enumerative Functions, University of Banja Luka (Bosnia and Herzegovina, 2017).
• Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A053123, A053128, A000973.

Programs

Magma

[Binomial(2*n-6,7): n in [7..40]]; // Vincenzo Librandi, Oct 07 2011

Maple

A053129:=n->binomial(2*n-6,7); seq(A053129(n), n=7..50); # Wesley Ivan Hurt, Nov 14 2013

Mathematica

Table[Binomial[2 n - 6, 7], {n, 7, 50}] (* Wesley Ivan Hurt, Nov 14 2013 *)

PARI

for(n=7, 50, print1(binomial(2*n-6,7), ", ")) \\ G. C. Greubel, Aug 26 2018

Formula

a(n) = binomial(2*n-6, 7) if n >= 7 else 0.

a(n) = -A053123(n,7), n >= 7; a(n) := 0, n=0..6, (eighth column of shifted Chebyshev's S-triangle, decreasing order).

a(n) = 8*A000973(n).

G.f.: (8+56*x+56*x^2+8*x^3)/(1-x)^8.

a(n) = (n-6)*(n-5)*(n-4)*(n-3)*(2*n-11)*(2*n-9)*(2*n-7)/315. - Wesley Ivan Hurt, Mar 25 2020

From Amiram Eldar, Oct 21 2022: (Start)

Sum_{n>=7} 1/a(n) = 777/5 - 224*log(2).

Sum_{n>=7} (-1)^(n+1)/a(n) = 441/10 - 14*Pi. (End)