A002678 Numerators of the Taylor coefficients of (e^x-1)^2. (Formerly M4321 N1810)

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

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

Also numerator of 2*Stirling2(n,2)/n!. - N. J. A. Sloane, Feb 14 2019

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. See Table, row m=2.

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

The sequences of rationals in the Salzer (1944) table are 1/A000142, A002678/A002679, A065974/A065975, A324003/A324004, A324005/A324006.

Sequence in context: A301731 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