OFFSET
0,2
COMMENTS
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
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294, Problem 7.15.
R. L. Graham, D. E. Knuth, O. Patashnik; Concrete Mathematics, Addison-Wesley, Reading (1994) 2nd Ed. Exercise 5.100.
G. Letac, Problèmes de probabilités, Presses Universitaires de France (1970), p. 14.
F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..200
T. Sillke, More information
D. Singmaster, Problem 79-16, Resistances in an n-Dimensional Cube, SIAM Review, 22 (1980) 504.
B. Sury, Sum of the reciprocals of the binomial coefficients, Europ. J. Combinatorics, 14 (1993), 351-353.
FORMULA
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)
G.f. for P(n): (2*log(1-z))/(-2+z). - Wouter Meeussen
P(n) = 2^(-n) * Sum_{k=1..n} (binomial(n,k) + 1)/k.
a(n) = numerator( A003149(n)/n! ). - G. C. Greubel, May 24 2021
EXAMPLE
MATHEMATICA
Numerator/@Table[Sum[1/Binomial[n, k], {k, 0, n}], {n, 0, 40}] (* Harvey P. Dale, Apr 21 2011 *)
PROG
(PARI) P=1; vector(30, n, numerator(P)+0*P=P/2/n*(n+1)+1) \\ M. F. Hasler, Jul 17 2012
(PARI) A046825(n)=numerator(sum(k=0, n, 1/binomial(n, k))) \\ M. F. Hasler, Jul 19 2012
(Magma) [Numerator((&+[1/Binomial(n, j): j in [0..n]])): n in [0..40]]; // G. C. Greubel, May 24 2021
(Sage) [numerator(sum(1/binomial(n, j) for j in (0..n))) for n in (0..40)] # G. C. Greubel, May 24 2021
CROSSREFS
KEYWORD
nonn,easy,frac,nice
AUTHOR
EXTENSIONS
References entries (Comtet, Graham et al., Letac, Nedemeyer) and Links entries (Singmaster, Sury) from Torsten.Sillke(AT)uni-bielefeld.de
STATUS
approved