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 A242023 Decimal expansion of Sum(n >= 1, (-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3)). 7
 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 G. C. Greubel, Table of n, a(n) for n = 0..5000 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}] RealDigits[32*Log[2] - 64/3, 10, 50][[1]] (* G. C. Greubel, Nov 23 2017 *) 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 (MAGMA) [32*Log(2) - 64/3]; // G. C. Greubel, Nov 23 2017 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 April 16 04:36 EDT 2021. Contains 343030 sequences. (Running on oeis4.)