A053131 Binomial coefficients C(2*n-8,9).

10, 220, 2002, 11440, 48620, 167960, 497420, 1307504, 3124550, 6906900, 14307150, 28048800, 52451256, 94143280, 163011640, 273438880, 445891810, 708930508, 1101716330, 1677106640, 2505433700, 3679075400, 5317936260, 7575968400, 10648873950, 14783142660

Offset: 9

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 = 9..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.
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45, 10,-1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A053123, A053130, A053133.

Programs

Magma

[Binomial(2*n-8,9): n in [9..40]]; // Vincenzo Librandi, Oct 07 2011

Mathematica

Binomial[2*Range[9,40]-8,9] (* Harvey P. Dale, Mar 19 2012 *)

PARI

for(n=9,50, print1(binomial(2*n-8,9), ", ")) \\ G. C. Greubel, Aug 26 2018

Formula

a(n) = binomial(2*n-8, 9) if n >= 9 else 0.

G.f.: (10+120*x+252*x^2+120*x^3+10*x^4)/(1-x)^10.

a(n) = 2*A053133(n).

a(n) = -A053123(n,9), n >= 9; a(n) := 0, n=0..8 (tenth column of shifted Chebyshev's S-triangle, decreasing order).

From Amiram Eldar, Oct 21 2022: (Start)

Sum_{n>=9} 1/a(n) = 223611/280 - 1152*log(2).

Sum_{n>=9} (-1)^(n+1)/a(n) = 72*log(2) - 13947/280. (End)