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%I #47 Feb 21 2022 01:17:07
%S 1,2,5,8,8,13,151,256,83,146,1433,2588,15341,28211,52235,19456,19345,
%T 36362,651745,6168632,1463914,2786599,122289917,233836352,140001721,
%U 268709146,774885169,1491969394,41711914513,80530073893
%N Numerator of Sum_{k=0..n} 1/binomial(n,k).
%C The term a(12)=15341 is divisible by 23^2. Is there another term a(n) divisible by the square of a prime p larger than n+1? - _M. F. Hasler_, Jul 17 2012
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294, Problem 7.15.
%D R. L. Graham, D. E. Knuth, O. Patashnik; Concrete Mathematics, Addison-Wesley, Reading (1994) 2nd Ed. Exercise 5.100.
%D G. Letac, Problèmes de probabilités, Presses Universitaires de France (1970), p. 14.
%D F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.
%H T. D. Noe, <a href="/A046825/b046825.txt">Table of n, a(n) for n = 0..200</a>
%H T. Mansour, <a href="http://www.arXiv.org/abs/math.CO/0104026">Gamma function, beta function and combinatorial identities</a>.
%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/sum_reciprocal_binomials">More information</a>
%H D. Singmaster, <a href="https://www.jstor.org/stable/2029816">Problem 79-16, Resistances in an n-Dimensional Cube</a>, SIAM Review, 22 (1980) 504.
%H B. Sury, <a href="https://doi.org/10.1006/eujc.1993.1038">Sum of the reciprocals of the binomial coefficients</a>, Europ. J. Combinatorics, 14 (1993), 351-353.
%F Let P(n) = (1/n) * Sum_{k=0..n-1} 1/binomial(n-1, k) = A046878(n)/A046879(n) = A046825(n-1)/(n*A046826(n-1)): { 0, 1, 1, 5/6, 2/3, 8/15, ...}. Then P(n) = 2^(-n) * Sum_{k=1..n} 2^k / k = 2^(-n+1) * Sum_{k odd} binomial(n, k)/k; P(0) = 0, P(n) = P(n-1)/2 + 1/n. - Torsten Sillke (Torsten.Sillke(AT)uni-bielefeld.de)
%F G.f. for P(n): (2*log(1-z))/(-2+z). - _Wouter Meeussen_
%F P(n) = 2^(-n) * Sum_{k=1..n} (binomial(n,k) + 1)/k.
%F a(n) = numerator( A003149(n)/n! ). - _G. C. Greubel_, May 24 2021
%e 1, 2, 5/2, 8/3, 8/3, 13/5, 151/60, 256/105, 83/35, 146/63, 1433/630, 2588/1155, 15341/6930, 28211/12870, 52235/24024, 19456/9009, 19345/9009, ... = A046825/A046826
%t Numerator/@Table[Sum[1/Binomial[n,k],{k,0,n}],{n,0,40}] (* _Harvey P. Dale_, Apr 21 2011 *)
%o (PARI) P=1;vector(30,n,numerator(P)+0*P=P/2/n*(n+1)+1) \\ _M. F. Hasler_, Jul 17 2012
%o (PARI) A046825(n)=numerator(sum(k=0,n,1/binomial(n,k))) \\ _M. F. Hasler_, Jul 19 2012
%o (Magma) [Numerator((&+[1/Binomial(n,j): j in [0..n]])): n in [0..40]]; // _G. C. Greubel_, May 24 2021
%o (Sage) [numerator(sum(1/binomial(n,j) for j in (0..n))) for n in (0..40)] # _G. C. Greubel_, May 24 2021
%Y Cf. A003149, A046826, A048211, A051389, A100512, A100513.
%K nonn,easy,frac,nice
%O 0,2
%A _N. J. A. Sloane_
%E References entries (Comtet, Graham et al., Letac, Nedemeyer) and Links entries (Singmaster, Sury) from Torsten.Sillke(AT)uni-bielefeld.de
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