A002422 Expansion of (1-4*x)^(5/2).

1, -10, 30, -20, -10, -12, -20, -40, -90, -220, -572, -1560, -4420, -12920, -38760, -118864, -371450, -1179900, -3801900, -12406200, -40940460, -136468200, -459029400, -1556708400, -5318753700, -18296512728, -63334082520

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

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

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
• N. J. A. Sloane, Notes on A984 and A2420-A2424

Crossrefs

Cf. A007054, A004001, A002420, A002421, A002423, A002424, A007272, A001622.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(5/2) )); // G. C. Greubel, Jul 03 2019

Maple

A002422 := n -> -(15/8)*4^n*GAMMA(n-5/2)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002422(n), n=0..26); # Peter Luschny, Dec 14 2015

Mathematica

CoefficientList[Series[(1-4x)^{5/2},{x,0,30}],x] (* Vincenzo Librandi, Jun 11 2012 *)

PARI

vector(30, n, n--; (-4)^n*binomial(5/2, n)) \\ G. C. Greubel, Jul 03 2019

Sage

[(-4)^n*binomial(5/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

Formula

a(n+3) = -2 * A007272(n).

a(n) = Sum_{m=0..n} binomial(n, m) * K_m(6), where K_m(x) = K_m(n, 2, x) is a Krawtchouk polynomial. - Alexander Barg (abarg(AT)research.bell-labs.com).

a(n) ~ -15/8*Pi^(-1/2)*n^(-7/2)*2^(2*n)*{1 + 35/8*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001

a(n) = -(15/8)*4^n*Gamma(n-5/2)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015

a(n) = (-4)^n*binomial(5/2, n). - Peter Luschny, Oct 22 2018

D-finite with recurrence: n*a(n) +2*(-2*n+7)*a(n-1)=0. - R. J. Mathar, Jan 16 2020

From Amiram Eldar, Mar 24 2022: (Start)

Sum_{n>=0} 1/a(n) = 32/45 - 14*Pi/(3^5*sqrt(3)).

Sum_{n>=0} (-1)^n/a(n) = 2144/1875 - 28*log(phi)/(5^4*sqrt(5)), where phi is the golden ratio (A001622). (End)