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%I M5058 N2188 #47 Mar 25 2022 05:14:40
%S 1,-18,126,-420,630,-252,-84,-72,-90,-140,-252,-504,-1092,-2520,-6120,
%T -15504,-40698,-110124,-305900,-869400,-2521260,-7443720,-22331160,
%U -67964400,-209556900,-653817528,-2062039896,-6567978928,-21111360840
%N Expansion of (1-4*x)^(9/2).
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D T. N. Thiele, Interpolationsrechnung. Teubner, Leipzig, 1909, p. 164.
%H Vincenzo Librandi, <a href="/A002424/b002424.txt">Table of n, a(n) for n = 0..1000</a>
%H Alexander Barg, <a href="https://arxiv.org/abs/2005.12995">Stolarsky's invariance principle for finite metric spaces</a>, arXiv:2005.12995 [math.CO], 2020.
%H N. J. A. Sloane, <a href="/A000984/a000984.pdf">Notes on A984 and A2420-A2424</a>.
%F a(n) = Sum_{m=0..n} binomial(n, m) * K_m(10), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg, abarg(AT)research.bell-labs.com.
%F a(n) = -(945/32)*4^n*Gamma(-9/2+n)/(sqrt(Pi)*Gamma(1+n)). - _Peter Luschny_, Dec 14 2015
%F a(n) = (-4)^n*binomial(9/2, n). - _G. C. Greubel_, Jul 03 2019
%F D-finite with recurrence: n*a(n) +2*(-2*n+11)*a(n-1)=0. - _R. J. Mathar_, Jan 16 2020
%F From _Amiram Eldar_, Mar 25 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 32/35 - 22*Pi/(3^7*sqrt(3)).
%F Sum_{n>=0} (-1)^n/a(n) = 1050752/984375 - 44*log(phi)/(5^6*sqrt(5)), where phi is the golden ratio (A001622). (End)
%p A002424 := n -> -(945/32)*4^n*GAMMA(-9/2+n)/(sqrt(Pi)*GAMMA(1+n)):
%p seq(A002424(n),n=0..28); # _Peter Luschny_, Dec 14 2015
%t CoefficientList[Series[(1-4x)^(9/2),{x,0,30}],x] (* _Harvey P. Dale_, Dec 27 2011 *)
%o (PARI) my(x='x+O('x^30)); Vec((1-4*x)^(9/2)) \\ _Altug Alkan_, Dec 14 2015
%o (PARI) vector(30, n, n--; (-4)^n*binomial(9/2, n)) \\ _G. C. Greubel_, Jul 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(9/2) )); // _G. C. Greubel_, Jul 03 2019
%o (Sage) [(-4)^n*binomial(9/2, n) for n in (0..30)] # _G. C. Greubel_, Jul 03 2019
%Y Cf. A001622, A002420, A002421, A002422, A002423, A004001, A007054, A007272.
%K sign,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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