This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A242023 #36 Dec 25 2023 17:35:44
%S A242023 8,4,7,3,7,6,4,4,4,5,8,4,9,1,6,5,6,8,0,1,8,0,9,4,5,5,3,3,2,8,3,1,6,8,
%T A242023 4,5,0,8,2,6,7,0,9,6,6,1,9,4,8,3,4,7,9,8,5,2,8,4,2,6,9,7,0,4,5,5,2,6,
%U A242023 2,5,6,9,6,9
%N A242023 Decimal expansion of Sum_{n >= 1} (-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3)).
%C A242023 Sum of terms of the inverse of Binomial(n,4) or A000332, for n>=4, with alternating signs.
%C A242023 In general the sums of Binomial coefficients of this form appear to have the form m*log(2) - r, where m is an integer and r is rational as below:
%C A242023 For Binomial(n,1): m = 1, r = 0. See A002162.
%C A242023 For Binomial(n,2): m = 4, r = 2. See A000217.
%C A242023 For Binomial(n,3): m = 12 r = 15/2. See A000292.
%C A242023 For Binomial(n,4): m = 32, r = 64/3. See A000332.
%C A242023 For Binomial(n,5): m = 80, r = 655/12. See A000389.
%C A242023 For Binomial(n,6): m = 192, r = 661/5. See A000579.
%C A242023 For Binomial(n,7): m = 448, r = 9289/30. See A000580.
%C A242023 For Binomial(n,8): m = 1024, r = 74432/105. See A000581.
%C A242023 This is generalized as follows:
%C A242023 m grows as A001787(k) = k*2^(k-1) for Binomial(n,k).
%C A242023 r * (k-1)! produces the integer sequence: a(k) = 0, 2, 15, 128, 1310, 15864, 222936, 3572736, where a(k+1)/a(k) approaches 2*k for large k.
%C A242023 Results are precise to 100 digits or more using Mathematica.
%H A242023 G. C. Greubel, <a href="/A242023/b242023.txt">Table of n, a(n) for n = 0..5000</a>
%F A242023 Equals 32*log(2) - 64/3.
%F A242023 Equals 32*(A259284-1). - _R. J. Mathar_, Jun 30 2021
%e A242023 0.8473764445849165680180945...
%t A242023 Sum[N[(-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3)), 150], {n, 1, Infinity}]
%t A242023 RealDigits[32*Log[2] - 64/3, 10, 50][[1]] (* _G. C. Greubel_, Nov 23 2017 *)
%o A242023 (PARI) 32*log(2) - 64/3 \\ _Michel Marcus_, Aug 13 2014
%o A242023 (PARI) sumalt(n=1, (-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3))) \\ _Michel Marcus_, Aug 14 2014
%o A242023 (Magma) [32*Log(2) - 64/3]; // _G. C. Greubel_, Nov 23 2017
%Y A242023 Cf. A000332, A000217, A000292, A242024, A002162, A001787.
%K A242023 nonn,cons,easy
%O A242023 0,1
%A A242023 _Richard R. Forberg_, Aug 11 2014