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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-Fran├žois Alcover, Apr 03 2014

REFERENCES

H. E. Salzer, Tables of coefficients for differences in terms of their derivatives, Journal of Mathematics and Physics, 23 (1944), 210-212.

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

Index entries for sequences related to Bernoulli numbers.

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

N. J. A. Sloane.

EXTENSIONS

More terms from David Broadhurst, Jan 19 2006

More terms from Stefan Steinerberger, Apr 04 2006

STATUS

approved

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Last modified April 21 02:07 EDT 2014. Contains 240824 sequences.