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Expansion of (1-4*x)^(7/2).
(Formerly M4934 N2114)
12

%I M4934 N2114 #51 Mar 24 2022 08:05:10

%S 1,-14,70,-140,70,28,28,40,70,140,308,728,1820,4760,12920,36176,

%T 104006,305900,917700,2801400,8684340,27293640,86843400,279409200,

%U 908079900,2978502072,9851968392,32839894640

%N Expansion of (1-4*x)^(7/2).

%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="/A002423/b002423.txt">Table of n, a(n) for n = 0..1000</a>

%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(8), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com)

%F a(n) ~ 105*4^(n-2)/(sqrt(Pi)*n^(9/2)). - _Vaclav Kotesovec_, Jul 28 2013

%F a(n) = (105/16)*4^n*Gamma(-7/2+n)/(sqrt(Pi)*Gamma(1+n)). - _Peter Luschny_, Dec 14 2015

%F a(n) = (-4)^n * binomial(7/2, n). - _G. C. Greubel_, Jul 03 2019

%F D-finite with recurrence: n*a(n) +2*(-2*n+9)*a(n-1)=0. - _R. J. Mathar_, Jan 16 2020

%F From _Amiram Eldar_, Mar 24 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 36/35 + 2*Pi/(3^4*sqrt(3)).

%F Sum_{n>=0} (-1)^n/a(n) = 23932/21875 - 36*log(phi)/(5^5*sqrt(5)), where phi is the golden ratio (A001622). (End)

%p A002423 := n -> (105/16)*4^n*GAMMA(-7/2+n)/(sqrt(Pi)*GAMMA(1+n)):

%p seq(A002423(n), n=0..27); # _Peter Luschny_, Dec 14 2015

%t CoefficientList[Series[(1-4*x)^(7/2),{x,0,30}],x] (* _Jean-François Alcover_, Mar 21 2011 *)

%t Table[(4^(-1+x) Pochhammer[-(7/2),-1+x])/Pochhammer[1,-1+x],{x,30}] (* _Harvey P. Dale_, Jul 13 2011 *)

%o (PARI) vector(30, n, n--; (-4)^n*binomial(7/2, n)) \\ _G. C. Greubel_, Jul 03 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(7/2) )); // _G. C. Greubel_, Jul 03 2019

%o (Sage) [(-4)^n*binomial(7/2, n) for n in (0..30)] # _G. C. Greubel_, Jul 03 2019

%Y Cf. A007054, A004001, A002420, A002421, A002422, A002424, A007272, A001622.

%K sign,easy,nice

%O 0,2

%A _N. J. A. Sloane_