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%I #45 Feb 10 2024 11:36:37
%S 1,101,5151,176851,4598126,96560646,1705904746,26075972546,
%T 352025629371,4263421511271,46897636623981,473239787751081,
%U 4416904685676756,38393094575497956,312629484400483356,2396826047070372396,17376988841260199871,119594570260437846171
%N a(n) = binomial coefficient C(n,100).
%C 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
%H G. C. Greubel, <a href="/A017764/b017764.txt">Table of n, a(n) for n = 100..1100</a>
%F G.f.: x^100/(1 - x)^101. - _Ilya Gutkovskiy_, Mar 21 2016
%F E.g.f.: x^100 * exp(x)/(100)!. - _G. C. Greubel_, Nov 24 2017
%F From _Amiram Eldar_, Dec 20 2020: (Start)
%F Sum_{n>=100} 1/a(n) = 100/99.
%F Sum_{n>=100} (-1)^n/a(n) = A001787(100)*log(2) - A242091(100)/99! = 63382530011411470074835160268800*log(2) - 1914409165727592211172313915606932788039791776845041612575266508424929 / 43575234518570298227833630584570189723 = 0.9902877001... (End)
%t Table[Binomial[n, 100], {n, 100, 5!}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 25 2008 *)
%o (Sage) [binomial(n, 100) for n in range(100,115)] # _Zerinvary Lajos_, May 23 2009
%o (PARI) a(n)=binomial(n,100) \\ _Charles R Greathouse IV_, Jun 28 2012
%o (Python)
%o A017764_list, m = [], [1]*101
%o for _ in range(10**2):
%o A017764_list.append(m[-1])
%o for i in range(100):
%o m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016
%o (Magma) [Binomial(n,100): n in [100..130]]; // _G. C. Greubel_, Nov 24 2017
%Y 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).
%Y Cf. A001787, A242091.
%K nonn,easy
%O 100,2
%A _N. J. A. Sloane_
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