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Search: id:A002678
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| A002678 |
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Numerators of the Taylor coefficients of (e^x-1)^2.
(Formerly M4321 N1810)
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+0
2
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| 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; listen)
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OFFSET
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2,3
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COMMENT
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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 (D.Broadhurst(AT)open.ac.uk), Jan 19 2006
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
H. E. Salzer, Tables of coefficients for differences in terms of their derivatives, Journal of Mathematics and Physics, 23 (1944), 210-212.
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LINKS
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T. D. Noe, Table of n, a(n) for n=2..300
Index entries for sequences related to Bernoulli numbers.
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FORMULA
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a(n) is the numerator of (2^n-2)/n! with generating function (e^x-1)^2 - David Broadhurst (D.Broadhurst(AT)open.ac.uk), Jan 19 2006
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MATHEMATICA
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Table[Numerator[Coefficient[Series[(E^x - 1)^2, {x, 0, 60}], x^n]], {n, 2, 60}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 04 2006
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PROGRAM
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(PARI) print(vector(30, n, numerator((2^n-2)/n!))) (Broadhurst)
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CROSSREFS
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Cf. A002679.
Sequence in context: A146996 A083994 A084181 this_sequence A147482 A050402 A027643
Adjacent sequences: A002675 A002676 A002677 this_sequence A002679 A002680 A002681
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KEYWORD
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nonn,frac
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from David Broadhurst (D.Broadhurst(AT)open.ac.uk), Jan 19 2006
More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 04 2006
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