OFFSET
0,2
FORMULA
G.f.: A(x) = sqrt( (1+2x)/(1-2x)^3 ).
a(n) = Sum_{k=0..[(n+1)/2]} C(n+1, k)*(n+1-2k)^3/(n+1).
a(n) = A107233(n)/(n+1).
Self-convolution equals A014477.
E.g.f.: ((1 + 4*x)*BesselI(0, 2*x) + 4*x*BesselI(1, 2*x)). - Peter Luschny, Aug 26 2012
a(n) = (-2)^n*hypergeom([-n,3/2], [1], 2). - Peter Luschny, Apr 26 2016
D-finite with recurrence: (n+1)*a(n+1) = 4*a(n) + 4*n*a(n-1). - Vladimir Reshetnikov, Oct 10 2016
a(n) ~ 2^(n + 3/2) * sqrt(n/Pi). - Vaclav Kotesovec, Oct 11 2016
From Peter Bala, Mar 31 2024: (Start)
a(n) = (2^n) * Sum_{k = 0..n} (-1)^(n+k)*binomial(1/2, k)*binomial(-3/2, n-k).
a(n) = (2^n) * Sum_{k = 0..n} (2^k)*binomial(n, k)*binomial(1/2, k).
a(n) = (2^n)* Sum_{k = 0..n} binomial(n, k)*binomial(k+1/2, n). See A008288.
a(n) = (2*n + 1)!/(2^n * n!^2) * hypergeom([-n, -1/2], [-n-1/2], -1).
a(n) = 2^n * hypergeom([-n, -1/2], [1], 2).
a(n) = (-1/2)^n * binomial(2*n, n)/(1 - 2*n) * hypergeom([-n, 3/2], [-n+3/2], -1).(End)
MAPLE
seq(simplify((-2)^n*hypergeom([-n, 3/2], [1], 2)), n=0..29); # Peter Luschny, Apr 26 2016
MATHEMATICA
CoefficientList[Series[Sqrt[(1+2x)/(1-2x)^3], {x, 0, 30}], x] (* Harvey P. Dale, Jan 04 2016 *)
PROG
(PARI) a(n)=sum(k=0, floor((n+1)/2), binomial(n+1, k)*(n+1-2*k)^3)/(n+1)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 14 2008
STATUS
approved