A242023 - OEIS

%I #36 Dec 25 2023 17:35:44

%S 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 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 2,5,6,9,6,9

%N Decimal expansion of Sum_{n >= 1} (-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3)).

%C Sum of terms of the inverse of Binomial(n,4) or A000332, for n>=4, with alternating signs.

%C 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 For Binomial(n,1):  m = 1, r = 0. See A002162.

%C For Binomial(n,2):  m = 4, r = 2. See A000217.

%C For Binomial(n,3):  m = 12 r = 15/2. See A000292.

%C For Binomial(n,4):  m = 32, r = 64/3. See A000332.

%C For Binomial(n,5):  m = 80, r = 655/12. See A000389.

%C For Binomial(n,6):  m = 192, r = 661/5. See A000579.

%C For Binomial(n,7):  m = 448, r = 9289/30. See A000580.

%C For Binomial(n,8):  m = 1024, r = 74432/105. See A000581.

%C This is generalized as follows:

%C m grows as A001787(k) = k*2^(k-1) for Binomial(n,k).

%C 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 Results are precise to 100 digits or more using Mathematica.

%H G. C. Greubel, <a href="/A242023/b242023.txt">Table of n, a(n) for n = 0..5000</a>

%F Equals 32*log(2) - 64/3.

%F Equals 32*(A259284-1). - _R. J. Mathar_, Jun 30 2021

%e 0.8473764445849165680180945...

%t Sum[N[(-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3)), 150], {n, 1, Infinity}]

%t RealDigits[32*Log[2] - 64/3, 10, 50][[1]] (* _G. C. Greubel_, Nov 23 2017 *)

%o (PARI) 32*log(2) - 64/3 \\ _Michel Marcus_, Aug 13 2014

%o (PARI) sumalt(n=1, (-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3))) \\ _Michel Marcus_, Aug 14 2014

%o (Magma) [32*Log(2) - 64/3]; // _G. C. Greubel_, Nov 23 2017

%Y Cf. A000332, A000217, A000292, A242024, A002162, A001787.

%K nonn,cons,easy

%O 0,1

%A _Richard R. Forberg_, Aug 11 2014