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A002675
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Numerators of coefficients for central differences M_{4}^(2*n).
(Formerly M5035 N2173)
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10
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1, 1, 1, 17, 31, 1, 5461, 257, 73, 1271, 60787, 241, 22369621, 617093, 49981, 16843009, 5726623061, 7957, 91625968981, 61681, 231927781, 50991843607, 499069107643, 4043309297, 1100586419201, 5664905191661, 1672180312771
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OFFSET
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2,4
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COMMENTS
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Numerators in the expansion of (2*sinh(x/2))^4 = x^4 + (1/6)*x^6 + (1/80)*x^8 + (17/30240)*x^10 + ....
Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^4 leads to a formula for the fourth central differences: f(x+2) - 4*f(x+1) + 6*f(x) - 4*f(x-1) + f(x-2) = (2*sinh(D/2))^4(f(x)) = D^4(f(x)) + (1/6)*D^6(f(x)) + (1/80)* D^8(f(x)) + (17/30240)*D^10(f(x)) + ..., where D denotes the differential operator d/dx. (End)
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REFERENCES
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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).
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LINKS
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MAPLE
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gf := (sinh(2*sqrt(x)) - 2*sinh(sqrt(x)))*sqrt(x):
ser := series(gf, x, 40): seq(numer(coeff(ser, x, n)), n=2..28); # Peter Luschny, Oct 05 2019
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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