A002423 Expansion of (1-4*x)^(7/2). (Formerly M4934 N2114)

1, -14, 70, -140, 70, 28, 28, 40, 70, 140, 308, 728, 1820, 4760, 12920, 36176, 104006, 305900, 917700, 2801400, 8684340, 27293640, 86843400, 279409200, 908079900, 2978502072, 9851968392, 32839894640

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

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

Formula

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)

a(n) ~ 105*4^(n-2)/(sqrt(Pi)*n^(9/2)). - Vaclav Kotesovec, Jul 28 2013

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

a(n) = (-4)^n * binomial(7/2, n). - G. C. Greubel, Jul 03 2019

D-finite with recurrence: n*a(n) +2*(-2*n+9)*a(n-1)=0. - R. J. Mathar, Jan 16 2020

From Amiram Eldar, Mar 24 2022: (Start)

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

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

Maple

A002423 := n -> (105/16)*4^n*GAMMA(-7/2+n)/(sqrt(Pi)*GAMMA(1+n)):
seq(A002423(n), n=0..27); # Peter Luschny, Dec 14 2015

Mathematica

CoefficientList[Series[(1-4*x)^(7/2), {x, 0, 30}], x] (* Jean-François Alcover, Mar 21 2011 *)
Table[(4^(-1+x) Pochhammer[-(7/2), -1+x])/Pochhammer[1, -1+x], {x, 30}] (* Harvey P. Dale, Jul 13 2011 *)

Prog

(PARI) vector(30, n, n--; (-4)^n*binomial(7/2, n)) \\ G. C. Greubel, Jul 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(7/2) )); // G. C. Greubel, Jul 03 2019
(Sage) [(-4)^n*binomial(7/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

Crossrefs

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

Sequence in context: A008354 A051879 A236157 * A212751 A212749 A201106

Adjacent sequences: A002420 A002421 A002422 * A002424 A002425 A002426

Keyword

sign,easy,nice

Author

N. J. A. Sloane

Status

approved