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 A002678 Numerators of the Taylor coefficients of (e^x-1)^2. (Formerly M4321 N1810) 2
 1, 1, 7, 1, 31, 1, 127, 17, 73, 31, 2047, 1, 8191, 5461, 4681, 257, 131071, 73, 524287, 1271, 42799, 60787, 8388607, 241, 33554431, 22369621, 19173961, 617093, 536870911, 49981, 2147483647, 16843009, 53353631, 5726623061, 1108378657 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,3 COMMENTS 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 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 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n=2..300 H. E. Salzer, Tables of coefficients for differences in terms of the derivatives, Journal of Mathematics and Physics, 23 (1944), 210-212. FORMULA a(n) is the numerator of (2^n-2)/n! with generating function (e^x-1)^2. - David Broadhurst, Jan 19 2006 MATHEMATICA Table[Numerator[Coefficient[Series[(E^x - 1)^2, {x, 0, 60}], x^n]], {n, 2, 60}] (* Stefan Steinerberger, Apr 04 2006 *) PROG (PARI) print(vector(30, n, numerator((2^n-2)/n!))) \\ David Broadhurst, Jan 19 2006 CROSSREFS Cf. A002679. Sequence in context: A083994 A228498 A084181 * A147482 A171770 A050402 Adjacent sequences:  A002675 A002676 A002677 * A002679 A002680 A002681 KEYWORD nonn,frac AUTHOR EXTENSIONS More terms from David Broadhurst, Jan 19 2006 More terms from Stefan Steinerberger, Apr 04 2006 STATUS approved

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