OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..500
FORMULA
G.f.: 1 / ((1 - 4*x) * sqrt(1 + 4*x)).
D-finite with recurrence: n*a(n) = 2 * a(n-1) + 8 * (2*n - 1) * a(n-2).
a(n) = 4^n * Sum_{k=0, n} binomial( 2*k, k) / (-4)^k.
a(n) = A003148(n) * 2^n / n!. - Michael Somos, Mar 17 2011
Asymptotics: a(n) ~ 4^n / sqrt(2).
G.f.: y = A(x) satisfies 0 = (16*x^2 - 1) * y' + (24*x + 2) * y and 0 = y'^3 + 8 * y'^2 * y^3 + 216 * y^5 - 256 * y^7.
G.f.: 1/((1-4*x)*sqrt(1+4*x)) = 1/(1-4*x+2*x*(1-4*x)/G(0)) ; G(k) = 1 + x/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 21 2011
From Vladimir Reshetnikov, Nov 01 2015: (Start)
a(n) = 2^(n-1)*(sqrt(2)*2^n + (-1)^n*(2n+1)!!*hypergeom([1,n+3/2], [n+2], -1)/(n+1)!.
a(n) = 2^n*(2*n+1)!!*hypergeom([-n,1/2], [3/2], 2)/n!. (End)
a(n+1) = 4*a(n) - (-1)^(n)*binomial(2n+2,n+1). - G. C. Greubel, Nov 02 2015
EXAMPLE
G.f. = 1 + 2*x + 14*x^2 + 36*x^3 + 214*x^4 + 604*x^5 + 3340*x^6 + 9928*x^7 + ...
MATHEMATICA
CoefficientList[Series[1/((1-4x)Sqrt[1+4x]), {x, 0, 30}], x] (* Harvey P. Dale, Oct 14 2013 *)
Table[2^n (2n+1)!! Hypergeometric2F1[-n, 1/2, 3/2, 2]/n!, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
RecurrenceTable[{a[n+1] == 4*a[n] - (-1)^(n)*binom[2n+2, n+1], a[0]==1}, a, {n, 0, 100}] (* G. C. Greubel, Nov 02 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, 4^n * sum( k=0, n, binomial(2*k, k) / (-4)^k))};
(PARI) x='x+O('x^50); Vec(1/((1-4*x)*sqrt(1+4*x))) \\ Altug Alkan, Nov 02 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 18 2004
STATUS
approved