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A002422
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Expansion of (1-4*x)^(5/2).
(Formerly M4692 N2003)
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13
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1, -10, 30, -20, -10, -12, -20, -40, -90, -220, -572, -1560, -4420, -12920, -38760, -118864, -371450, -1179900, -3801900, -12406200, -40940460, -136468200, -459029400, -1556708400, -5318753700, -18296512728, -63334082520
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OFFSET
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0,2
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REFERENCES
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A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 55.
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).
T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.
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LINKS
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FORMULA
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a(n) = Sum_{m=0..n} binomial(n, m) * K_m(6), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com).
a(n) ~ -15/8*Pi^(-1/2)*n^(-7/2)*2^(2*n)*{1 + 35/8*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
a(n) = -(15/8)*4^n*Gamma(n-5/2)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
D-finite with recurrence: n*a(n) +2*(-2*n+7)*a(n-1)=0. - R. J. Mathar, Jan 16 2020
Sum_{n>=0} 1/a(n) = 32/45 - 14*Pi/(3^5*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 2144/1875 - 28*log(phi)/(5^4*sqrt(5)), where phi is the golden ratio (A001622). (End)
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MAPLE
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A002422 := n -> -(15/8)*4^n*GAMMA(n-5/2)/(sqrt(Pi)*GAMMA(1+n)):
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MATHEMATICA
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CoefficientList[Series[(1-4x)^{5/2}, {x, 0, 30}], x] (* Vincenzo Librandi, Jun 11 2012 *)
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PROG
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(PARI) vector(30, n, n--; (-4)^n*binomial(5/2, n)) \\ G. C. Greubel, Jul 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(5/2) )); // G. C. Greubel, Jul 03 2019
(Sage) [(-4)^n*binomial(5/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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