A242023 Decimal expansion of Sum_{n >= 1} (-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3)).

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, 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, 2, 5, 6, 9, 6, 9

Offset: 0

Author

Richard R. Forberg, Aug 11 2014

Keywords

• nonn
• cons
• easy

Comments

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

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:

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

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

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

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

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

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

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

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

This is generalized as follows:

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

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.

Results are precise to 100 digits or more using Mathematica.

Examples

			0.8473764445849165680180945...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

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

Programs

Magma

[32*Log(2) - 64/3]; // G. C. Greubel, Nov 23 2017

Mathematica

Sum[N[(-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3)), 150], {n, 1, Infinity}]
RealDigits[32*Log[2] - 64/3, 10, 50][[1]] (* G. C. Greubel, Nov 23 2017 *)

PARI

32*log(2) - 64/3 \\ Michel Marcus, Aug 13 2014

PARI

sumalt(n=1, (-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3))) \\ Michel Marcus, Aug 14 2014

Formula

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

Equals 32*(A259284-1). - R. J. Mathar, Jun 30 2021