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A002423
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Expansion of (1-4*x)^(7/2).
(Formerly M4934 N2114)
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12
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1, -14, 70, -140, 70, 28, 28, 40, 70, 140, 308, 728, 1820, 4760, 12920, 36176, 104006, 305900, 917700, 2801400, 8684340, 27293640, 86843400, 279409200, 908079900, 2978502072, 9851968392, 32839894640
(list;
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listen;
history;
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internal format)
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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(8), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com)
a(n) = (105/16)*4^n*Gamma(-7/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
D-finite with recurrence: n*a(n) +2*(-2*n+9)*a(n-1)=0. - R. J. Mathar, Jan 16 2020
Sum_{n>=0} 1/a(n) = 36/35 + 2*Pi/(3^4*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 23932/21875 - 36*log(phi)/(5^5*sqrt(5)), where phi is the golden ratio (A001622). (End)
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MAPLE
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A002423 := n -> (105/16)*4^n*GAMMA(-7/2+n)/(sqrt(Pi)*GAMMA(1+n)):
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MATHEMATICA
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Table[(4^(-1+x) Pochhammer[-(7/2), -1+x])/Pochhammer[1, -1+x], {x, 30}] (* Harvey P. Dale, Jul 13 2011 *)
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PROG
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(PARI) vector(30, n, n--; (-4)^n*binomial(7/2, n)) \\ G. C. Greubel, Jul 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(7/2) )); // G. C. Greubel, Jul 03 2019
(Sage) [(-4)^n*binomial(7/2, 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,easy,nice
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AUTHOR
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STATUS
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approved
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