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%I M4321 N1810 #38 Feb 15 2019 13:55:29
%S 1,1,7,1,31,1,127,17,73,31,2047,1,8191,5461,4681,257,131071,73,524287,
%T 1271,42799,60787,8388607,241,33554431,22369621,19173961,617093,
%U 536870911,49981,2147483647,16843009,53353631,5726623061,1108378657
%N Numerators of the Taylor coefficients of (e^x-1)^2.
%C In 1929, Phillip Morse showed that a potential energy function of the form (e^x-1)^2 leads to a soluble Schroedinger equation. The numerators of its Taylor coefficients contain the Mersenne primes greater than 3. - _David Broadhurst_, Jan 19 2006
%C The integral f(z) = int((exp(z*exp(-y^2))-1)^2, {y, -infinity, infinity}) can be computed as sum(sqrt(Pi/k)*A002678(k)*(z^k/A002679(k)), {k, 1, infinity}). - _Jean-François Alcover_, Apr 03 2014
%C Also numerator of 2*Stirling2(n,2)/n!. - _N. J. A. Sloane_, Feb 14 2019
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002678/b002678.txt">Table of n, a(n) for n=2..300</a>
%H H. E. Salzer, <a href="https://doi.org/10.1002/sapm1944231210">Tables of coefficients for differences in terms of the derivatives</a>, Journal of Mathematics and Physics, 23 (1944), 210-212. See Table, row m=2.
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F a(n) is the numerator of (2^n-2)/n! with generating function (e^x-1)^2. - _David Broadhurst_, Jan 19 2006
%t Table[Numerator[Coefficient[Series[(E^x - 1)^2, {x, 0, 60}], x^n]], {n, 2, 60}] (* _Stefan Steinerberger_, Apr 04 2006 *)
%o (PARI) print(vector(30,n,numerator((2^n-2)/n!))) \\ _David Broadhurst_, Jan 19 2006
%Y The sequences of rationals in the Salzer (1944) table are 1/A000142, A002678/A002679, A065974/A065975, A324003/A324004, A324005/A324006.
%K nonn,frac
%O 2,3
%A _N. J. A. Sloane_
%E More terms from _David Broadhurst_, Jan 19 2006
%E More terms from _Stefan Steinerberger_, Apr 04 2006
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