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A242023 Decimal expansion of Sum(n >= 1, (-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3)). 6
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

LINKS

Table of n, a(n) for n=0..73.

FORMULA

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

EXAMPLE

0.8473764445849165680180945...

MATHEMATICA

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

PROG

(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

CROSSREFS

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

Sequence in context: A096427 A176453 A257775 * A249415 A021122 A110233

Adjacent sequences:  A242020 A242021 A242022 * A242024 A242025 A242026

KEYWORD

nonn,cons

AUTHOR

Richard R. Forberg, Aug 11 2014

STATUS

approved

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Last modified May 23 16:52 EDT 2017. Contains 286925 sequences.