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

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

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

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

Crossrefs

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

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-4*x)^(7/2) )); // G. C. Greubel, Jul 03 2019

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 *)

PARI

vector(30, n, n--; (-4)^n*binomial(7/2, n)) \\ G. C. Greubel, Jul 03 2019

Sage

[(-4)^n*binomial(7/2, n) for n in (0..30)] # G. C. Greubel, Jul 03 2019

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)