A017764 a(n) = binomial coefficient C(n,100).

1, 101, 5151, 176851, 4598126, 96560646, 1705904746, 26075972546, 352025629371, 4263421511271, 46897636623981, 473239787751081, 4416904685676756, 38393094575497956, 312629484400483356, 2396826047070372396, 17376988841260199871, 119594570260437846171

Offset: 100

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

More generally, the ordinary generating function for the binomial coefficients C(n,k) is x^k/(1 - x)^(k+1). - Ilya Gutkovskiy, Mar 21 2016

Links

• G. C. Greubel, Table of n, a(n) for n = 100..1100

Crossrefs

Cf. similar sequences of the binomial coefficients C(n,k): A000012 (k = 0), A001477 (k = 1), A000217 (k = 2), A000292 (k = 3), A000332 (k = 4), A000389 (k = 5), A000579-A000582 (k = 6..9) A001287 (k = 10), A001288 (k = 11), A010965-A011001 (k = 12..48), A017713-A017763 (k = 49..99), this sequence (k = 100).

Cf. A001787, A242091.

Programs

Magma

[Binomial(n,100): n in [100..130]]; // G. C. Greubel, Nov 24 2017

Mathematica

Table[Binomial[n, 100], {n, 100, 5!}] (* Vladimir Joseph Stephan Orlovsky, Sep 25 2008 *)

PARI

a(n)=binomial(n,100) \\ Charles R Greathouse IV, Jun 28 2012

Python

A017764_list, m = [], [1]*101
for _ in range(10**2):
    A017764_list.append(m[-1])
    for i in range(100):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

Sage

[binomial(n, 100) for n in range(100,115)] # Zerinvary Lajos, May 23 2009

Formula

G.f.: x^100/(1 - x)^101. - Ilya Gutkovskiy, Mar 21 2016

E.g.f.: x^100 * exp(x)/(100)!. - G. C. Greubel, Nov 24 2017

From Amiram Eldar, Dec 20 2020: (Start)

Sum_{n>=100} 1/a(n) = 100/99.

Sum_{n>=100} (-1)^n/a(n) = A001787(100)*log(2) - A242091(100)/99! = 63382530011411470074835160268800*log(2) - 1914409165727592211172313915606932788039791776845041612575266508424929 / 43575234518570298227833630584570189723 = 0.9902877001... (End)