A091520 Expansion of 1 / ((1 - 4*x) * sqrt(1 + 4*x)) in powers of x.

1, 2, 14, 36, 214, 604, 3340, 9928, 52582, 161708, 831588, 2620920, 13187836, 42350744, 209519576, 682960784, 3332923526, 10998087884, 53067486836, 176924683544, 845545262996, 2843923177544, 13479791673896, 45685735967984

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

Sequence in context: A004117 A330672 A135706 * A338327 A322226 A218546

Adjacent sequences: A091517 A091518 A091519 * A091521 A091522 A091523

Keyword

nonn

Author

Michael Somos, Jan 18 2004

Status

approved