

A002678


Numerators of the Taylor coefficients of (e^x1)^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
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OFFSET

2,3


COMMENTS

In 1929, Phillip Morse showed that a potential energy function of the form (e^x1)^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}).  JeanFrançois 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), 210212.
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^n2)/n! with generating function (e^x1)^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^n2)/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



