A210736 Expansion of (1 + sqrt( (1 + 2*x) / (1 - 2*x))) / 2 in powers of x.

1, 1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716, 3432, 6435, 12870, 24310, 48620, 92378, 184756, 352716, 705432, 1352078, 2704156, 5200300, 10400600, 20058300, 40116600, 77558760, 155117520, 300540195, 601080390, 1166803110, 2333606220, 4537567650

Offset

0,4

Comments

Hankel transform is period 4 sequence [ 1, 0, -1, 0, ...] A056594 and the Hankel transform of sequence omitting a(0) is the all 1s sequence A000012. This is the unique sequence with that property.

Series reversion of x*A(x) apparently yields x*A036765(-x). - R. J. Mathar, Sep 24 2012

a(n) is the number of length n words on {-1,1} such that the sum of any of its prefixes is always positive. Cf. A001405 where the sum of all prefixes is nonnegative. - Geoffrey Critzer, Jul 08 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 77.

Formula

G.f.: 2 * x / (-1 + 2*x + sqrt(1 - 4*x^2)).

G.f. A(x) satisfies A(x) = A(x)^2 - x / (1 - 2*x).

G.f. A(x) satisfies A( x / (1 + x^2) ) = 1 / (1 - x).

G.f. A(x) satisfies A(1/3) = (1 + sqrt(5))/2.

G.f. A(x) = 1 + x / (1 - 2*x + x / A(x)).

G.f. A(x) = 1 + x / (1 - x / (1 - x / (1 + x / A(x)))).

G.f. A(x) = 1 + x * A001405(x). a(n+1) = A001405(n).

Convolution inverse is A210628. Partial sums is A072100.

Binomial transform with offset 1 is A211278 with offset 1. a(n+2) * a(n) - a(n+1)^2 = A138350(n-1).

a(n) = (-1)^floor(n/2)*hypergeom2F1([1-n, -n],[1],-1). - Peter Luschny, Sep 01 2012

D-finite with recurrence: n*a(n) -2*a(n-1) +4*(2-n)*a(n-2)=0. - R. J. Mathar, Sep 14 2012

G.f. A(x) = 1 / (1 - x / (1 - x^2 / (1 - x^2 / (1 - x^2 / ...)))). - Michael Somos, Jan 02 2013

G.f.: 1/(1 - x*C(x)) where C(x) is the o.g.f. for A126120. - Geoffrey Critzer, Jul 08 2013

a(n) ~ 2^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 01 2014

G.f.: A(x) = 1 - x/(- 1 + x/A(-x)). - Arkadiusz Wesolowski, Feb 28 2014

From Tom Copeland, Nov 07 2014: (Start)

Setting a(0)=0 here, we have a signed version in A126930 and

O.g.f. G(x)=[-1+sqrt(1+4*x/(1-2x))]/2 = x + x^2 + 2 x^3 + ... = -C[-P(P(x,-1),-1)]= -C[-P(x,-2)] where C(x)= [1-sqrt(1-4*x)]/2= x + x^2 + 2 x^3 + ... = A000108(x) with inverse Cinv(x)=x*(1-x), and P(x,t)= x/(1 + t*x) with inverse P(x,-t).

These types of arrays are from linear fractional transformations of C(x). See A091867.

Ginv(x) = P[-Cinv(-x),2] = x*(1+x)/(1+2*x*(1-x))= (x+x^2)/(1+2(x+x^2)) (see A146559). (End)

Example

G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 20*x^7 + 35*x^8 + 70*x^9 + ...

Mathematica

nn=36; d=(1-(1-4x^2)^(1/2))/(2x^2); CoefficientList[Series[1/(1-x d), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 08 2013 *)
CoefficientList[Series[2 x / (-1 + 2 x + Sqrt[1 - 4 x^2]), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)

Prog

(PARI) {a(n) = if( n<1, n==0, binomial( n - 1, (n - 1)\2))};
(PARI) {a(n) = polcoeff( (1 + sqrt( (1 + 2*x) / (1 - 2*x) + x * O(x^n))) / 2, n)};

Crossrefs

Essentially the same as A001405.

Cf. A056594, A138350, A210628, A211278.

Cf. A000108, A091867, A146559, A126930.

Sequence in context: A056202 A001405 A126930 * A036557 A173125 A047131

Adjacent sequences: A210733 A210734 A210735 * A210737 A210738 A210739

Keyword

nonn

Author

Michael Somos, May 10 2012

Status

approved