A004984 a(n) = (2^n/n!)*Product_{k=0..n-1} (4*k - 1).

1, -2, -6, -28, -154, -924, -5852, -38456, -259578, -1788204, -12517428, -88759944, -636112932, -4599585816, -33511268088, -245749299312, -1812401082426, -13433090375628, -100001895018564, -747382583822952, -5605369378672140, -42173731515723720

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• sign
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A004981, A025749, A034256, A048779, A303538.

Programs

GAP

List([0..25],n->(2^n/Factorial(n))*Product([0..n-1],k->4*k-1)); # Muniru A Asiru, Apr 28 2018

Magma

[1] cat [2^n*(&*[4*k-1: k in [0..n-1]])/Factorial(n): n in [1..30]]; // G. C. Greubel, Aug 22 2019

Maple

seq(-(1/4)*8^n*GAMMA(n-1/4)/(GAMMA(3/4)*n!), n=0..30); # Robert Israel, Sep 29 2014

Mathematica

Table[8^n*Pochhammer[-1/4, n]/n!, {n,0,30}] (* G. C. Greubel, Aug 22 2019 *)
CoefficientList[Series[Surd[1-8x,4],{x,0,30}],x] (* Harvey P. Dale, Dec 08 2019 *)

Maxima

a(n):=-1/n*(if n=1 then (2)^n else sum(sum(binomial(k,j)* binomial(j,n-1-3*k+2*j)*(3/2)^(3*k-n+1-j)*(-1)^(n-1-3*k)*(1/4)^(k-j),j,0,k)*binomial(k+n-1,n-1)*(2)^n,k,1,n-1)); /* Vladimir Kruchinin, Sep 14 2010 */

Maxima

a(n):=binomial(1/4,n)*(-8)^n; /* Tani Akinari, Sep 28 2014 */

PARI

for(n=0,28,print1(2^n/n!*prod(k=0,n-1,(4*k-1)),","))

Sage

[8^n*rising_factorial(-1/4, n)/factorial(n) for n in (0..30)] # G. C. Greubel, Aug 22 2019

Formula

G.f.: (1 - 8*x)^(1/4).

a(n) ~ -1/4*Gamma(3/4)^-1*n^(-5/4)*2^(3*n)*{1 + 5/32*n^-1 + ...}

a(n) = -1/n*Sum_{k=1..n-1} 2^n*binomial(k+n-1, n-1) * ( Sum_{j=0..k} binomial(k,j)*binomial(j,n-1-3*k+2*j)*(3/2)^(3*k-n+1-j)*(-1)^(n-1-3*k)*(1/4)^(k-j) ), n>1. - Vladimir Kruchinin, Sep 14 2010

a(n) = 8^n*Pochhammer(-1/4, n)/n! = -(1/4)*8^n*Gamma(n-1/4)/(Gamma(3/4)*n!). - Robert Israel, Sep 29 2014

D-finite with recurrence: n*a(n) +2*(-4*n+5)*a(n-1)=0. - R. J. Mathar, Jan 16 2020

Extensions

More terms from Jason Earls, Dec 04 2001