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

100,2

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

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)

Mathematica

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

Prog

(Sage) [binomial(n, 100) for n in range(100, 115)] # Zerinvary Lajos, May 23 2009
(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
(Magma) [Binomial(n, 100): n in [100..130]]; // G. C. Greubel, Nov 24 2017

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.

Sequence in context: A027900 A166218 A167626 * A217677 A333689 A291990

Adjacent sequences: A017761 A017762 A017763 * A017765 A017766 A017767

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved