A002424 Expansion of (1-4*x)^(9/2). (Formerly M5058 N2188)

1, -18, 126, -420, 630, -252, -84, -72, -90, -140, -252, -504, -1092, -2520, -6120, -15504, -40698, -110124, -305900, -869400, -2521260, -7443720, -22331160, -67964400, -209556900, -653817528, -2062039896, -6567978928, -21111360840

Offset

0,2

References

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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Alexander Barg, Stolarsky's invariance principle for finite metric spaces, arXiv:2005.12995 [math.CO], 2020.

N. J. A. Sloane, Notes on A984 and A2420-A2424.

Formula

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.

a(n) = -(945/32)*4^n*Gamma(-9/2+n)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015

a(n) = (-4)^n*binomial(9/2, n). - G. C. Greubel, Jul 03 2019

D-finite with recurrence: n*a(n) +2*(-2*n+11)*a(n-1)=0. - R. J. Mathar, Jan 16 2020

From Amiram Eldar, Mar 25 2022: (Start)

Sum_{n>=0} 1/a(n) = 32/35 - 22*Pi/(3^7*sqrt(3)).

Sum_{n>=0} (-1)^n/a(n) = 1050752/984375 - 44*log(phi)/(5^6*sqrt(5)), where phi is the golden ratio (A001622). (End)

Maple

A002424 := n -> -(945/32)*4^n*GAMMA(-9/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002424(n), n=0..28); # Peter Luschny, Dec 14 2015

Mathematica

CoefficientList[Series[(1-4x)^(9/2), {x, 0, 30}], x] (* Harvey P. Dale, Dec 27 2011 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-4*x)^(9/2)) \\ Altug Alkan, Dec 14 2015
(PARI) vector(30, n, n--; (-4)^n*binomial(9/2, n)) \\ G. C. Greubel, Jul 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(9/2) )); // G. C. Greubel, Jul 03 2019
(Sage) [(-4)^n*binomial(9/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

Crossrefs

Cf. A001622, A002420, A002421, A002422, A002423, A004001, A007054, A007272.

Sequence in context: A077960 A107971 A292314 * A101378 A107417 A056125

Adjacent sequences: A002421 A002422 A002423 * A002425 A002426 A002427

Keyword

sign,easy,nice

Author

N. J. A. Sloane

Status

approved