A053127 Binomial coefficients C(2*n-4,5).

6, 56, 252, 792, 2002, 4368, 8568, 15504, 26334, 42504, 65780, 98280, 142506, 201376, 278256, 376992, 501942, 658008, 850668, 1086008, 1370754, 1712304, 2118760, 2598960, 3162510, 3819816, 4582116, 5461512, 6471002, 7624512

Offset: 5

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 = 5..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 Janjic, 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 sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A053123, A053132, A053126.

Programs

Haskell

a053127 = (* 2) . a053132  -- Reinhard Zumkeller, Mar 03 2015

Magma

[Binomial(2*n-4,5): n in [5..40]]; // Vincenzo Librandi, Oct 07 2011

Mathematica

Binomial[2Range[5,40]-4,5] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{6,56,252,792,2002,4368},30] (* Harvey P. Dale, Jun 03 2013 *)

PARI

for(n=5,50, print1(binomial(2*n-4,5), ", ")) \\ G. C. Greubel, Aug 26 2018

Formula

a(n) = binomial(2*n-4, 5) if n >= 5 else 0.

a(n) = -A053123(n,5), n >= 5; a(n) := 0, n=0..4 (sixth column of shifted Chebyshev's S-triangle, decreasing order).

G.f.: (6+20*x+6*x^2)/(1-x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6). - Harvey P. Dale, Jun 03 2013

E.g.f.: (-840 + 750*x - 330*x^2 + 95*x^3 - 20*x^4 + 4*x^5)*exp(x)/15. - G. C. Greubel, Aug 26 2018

a(n) = (2*n-8)*(2*n-7)*(2*n-6)*(2*n-5)*(2*n-4)/120. - Wesley Ivan Hurt, Mar 25 2020

From Amiram Eldar, Jan 03 2022: (Start)

Sum_{n>=5} 1/a(n) = 335/12 - 40*log(2).

Sum_{n>=5} (-1)^(n+1)/a(n) = 85/12 - 10*log(2). (End)