A053126 Binomial coefficients binomial(2*n-3,4).

5, 35, 126, 330, 715, 1365, 2380, 3876, 5985, 8855, 12650, 17550, 23751, 31465, 40920, 52360, 66045, 82251, 101270, 123410, 148995, 178365, 211876, 249900, 292825, 341055, 395010, 455126, 521855, 595665, 677040, 766480

Offset

4,1

Comments

Number of intersections of diagonals in the interior of regular (2n-3)-gon. - Philippe Deléham, Jun 07 2013

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 = 4..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 (5,-10,10,-5,1).

Formula

a(n) = binomial(2*n-3, 4) if n >= 4 else 0;

G.f.: (5+10*x+x^2)/(1-x)^5.

a(n) = A053123(n,4), n >= 4; a(n) = 0, n=0..3 (fifth column of shifted Chebyshev's S-triangle, decreasing order).

a(n) = A006561(2n-3). - Philippe Deléham, Jun 07 2013

E.g.f.: (90 - 84*x + 39*x^2 - 12*x^3 + 4*x^4)*exp(x)/6. - G. C. Greubel, Aug 26 2018

From Amiram Eldar, Jan 04 2022: (Start)

Sum_{n>=4} 1/a(n) = 34/3 - 16*log(2).

Sum_{n>=4} (-1)^n/a(n) = 2*Pi - 4*log(2) - 10/3. (End)

Mathematica

Table[Binomial[2*n-3, 4], {n, 4, 50}] (* G. C. Greubel, Aug 26 2018 *)

Prog

(Magma) [Binomial(2*n-3, 4): n in [4..40]]; // Vincenzo Librandi, Oct 07 2011
(PARI) for(n=4, 50, print1(binomial(2*n-3, 4), ", ")) \\ G. C. Greubel, Aug 26 2018

Crossrefs

Cf. A053123, A002492.

Sequence in context: A161199 A111877 A179337 * A096743 A026697 A000910

Adjacent sequences: A053123 A053124 A053125 * A053127 A053128 A053129

Keyword

nonn,easy

Author

Wolfdieter Lang

Status

approved