A002422 Expansion of (1-4*x)^(5/2). (Formerly M4692 N2003)

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

0,2

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

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

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)

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 *)

Prog

(PARI) vector(30, n, n--; (-4)^n*binomial(5/2, n)) \\ G. C. Greubel, Jul 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(5/2) )); // G. C. Greubel, Jul 03 2019
(Sage) [(-4)^n*binomial(5/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

Crossrefs

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

Sequence in context: A027979 A181102 A057456 * A031195 A034117 A104863

Adjacent sequences: A002419 A002420 A002421 * A002423 A002424 A002425

Keyword

sign

Author

N. J. A. Sloane

Status

approved