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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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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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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Notes on A984 and A2420-A2424
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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*4^(n-2)/(sqrt(Pi)*n^(9/2)). - Vaclav Kotesovec, Jul 28 2013
a(n) = (105/16)*4^n*Gamma(-7/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) = (-4)^n * binomial(7/2, n). - G. C. Greubel, Jul 03 2019
D-finite with recurrence: n*a(n) +2*(-2*n+9)*a(n-1)=0. - R. J. Mathar, Jan 16 2020
From Amiram Eldar, Mar 24 2022: (Start)
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)):
seq(A002423(n), n=0..27); # Peter Luschny, Dec 14 2015
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MATHEMATICA
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CoefficientList[Series[(1-4*x)^(7/2), {x, 0, 30}], x] (* Jean-François Alcover, Mar 21 2011 *)
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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Cf. A007054, A004001, A002420, A002421, A002422, A002424, A007272, A001622.
Sequence in context: A008354 A051879 A236157 * A212751 A212749 A201106
Adjacent sequences: A002420 A002421 A002422 * A002424 A002425 A002426
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KEYWORD
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sign,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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