OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
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*(if n=1 then (2)^n else 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>0. - 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
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 *)
PROG
(PARI) for(n=0, 28, print1(2^n/n!*prod(k=0, n-1, (4*k-1)), ", "))
(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)); (for ) /* Vladimir Kruchinin, Sep 14 2010 */
(Maxima) a(n):=binomial(1/4, n)*(-8)^n; /* Tani Akinari, Sep 28 2014 */
(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
(Sage) [8^n*rising_factorial(-1/4, n)/factorial(n) for n in (0..30)] # G. C. Greubel, Aug 22 2019
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Joe Keane (jgk(AT)jgk.org)
EXTENSIONS
More terms from Jason Earls, Dec 04 2001
STATUS
approved