A053128 Binomial coefficients C(2*n-5,6).

7, 84, 462, 1716, 5005, 12376, 27132, 54264, 100947, 177100, 296010, 475020, 736281, 1107568, 1623160, 2324784, 3262623, 4496388, 6096454, 8145060, 10737573, 13983816, 18009460, 22957480, 28989675, 36288252, 45057474, 55525372, 67945521, 82598880, 99795696

Offset: 6

Author

Wolfdieter Lang

Keywords

• nonn
• easy

Comments

a(n) = A053123(n,6), n >= 6; a(n) = 0, n=0..5, (seventh column of shifted Chebyshev's S-triangle, decreasing order).

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 = 6..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 (7,-21,35,-35,21,-7,1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A053123, A053127.

Programs

Magma

[Binomial(2*n-5,6): n in [6..40]]; // Vincenzo Librandi, Oct 07 2011

Mathematica

Table[Binomial[2*n-5,6], {n,6,50}] (* G. C. Greubel, Aug 26 2018 *)

PARI

for(n=6,50, print1(binomial(2*n-5,6), ", ")) \\ G. C. Greubel, Aug 26 2018

Formula

a(n) = binomial(2*n-5, 6) if n >= 6 else 0.

G.f.: (7+35*x+21*x^2+x^3)/(1-x)^7.

E.g.f.: (18900 - 16380*x + 6975*x^2 - 1935*x^3 + 390*x^4 - 60*x^5 + 8*x^6)*exp(x)/90. - G. C. Greubel, Aug 26 2018

a(n) = (n-5)*(n-4)*(n-3)*(2*n-9)*(2*n-7)*(2*n-5)/90. - Wesley Ivan Hurt, Mar 25 2020

From Amiram Eldar, Oct 21 2022: (Start)

Sum_{n>=6} 1/a(n) = 667/10 - 96*log(2).

Sum_{n>=6} (-1)^n/a(n) = 273/10 - 6*Pi - 12*log(2). (End)