A010973 a(n) = binomial(n,20).

1, 21, 231, 1771, 10626, 53130, 230230, 888030, 3108105, 10015005, 30045015, 84672315, 225792840, 573166440, 1391975640, 3247943160, 7307872110, 15905368710, 33578000610, 68923264410, 137846528820, 269128937220, 513791607420, 960566918220, 1761039350070

Offset: 20

Author

N. J. A. Sloane

Keywords

• nonn

Comments

In this sequence there are no primes. - Artur Jasinski, Dec 02 2007

Links

• T. D. Noe, Table of n, a(n) for n = 20..1000
• Milan Janjic, Two Enumerative Functions.
• Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Crossrefs

Pascal's triangle A007318 diagonal. - Zerinvary Lajos, Aug 04 2008

Cf. A001787, A242091.

Programs

Magma

[ Binomial(n,20): n in [20..80]]; // Vincenzo Librandi, Mar 26 2011

Maple

seq(binomial(n,20),n=20..40); # Zerinvary Lajos, Aug 04 2008

Mathematica

Table[Binomial[n,20],{n,20,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

PARI

for(n=20,50, print1(binomial(n,20), ", ")) \\ G. C. Greubel, Nov 23 2017

Formula

a(n+19) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)*(n+6)*(n+7)*(n+8)*(n+9)*(n+10)*(n+11)*(n+12)*(n+13)*(n+14)*(n+15)*(n+16)*(n+17)*(n+18)*(n+19)/20!. - Artur Jasinski, Dec 02 2007; R. J. Mathar, Jul 07 2009

G.f.: x^20/(1-x)^21. - Zerinvary Lajos, Aug 04 2008; R. J. Mathar, Jul 07 2009

a(n) = n/(n-20) * a(n-1), n > 20. - Vincenzo Librandi, Mar 26 2011

From Amiram Eldar, Dec 11 2020: (Start)

Sum_{n>=20} 1/a(n) = 20/19.

Sum_{n>=20} (-1)^n/a(n) = A001787(20)*log(2) - A242091(20)/19! = 10485760*log(2) - 21149710469086/2909907 = 0.9562240549... (End)

Extensions

Some formulas adjusted to the offset by R. J. Mathar, Jul 07 2009