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

Author

Michael Somos, May 10 2012

Keywords

• nonn

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

Examples

			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 + ...
		

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).
• Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 25.
• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; page 77.

Crossrefs

Essentially the same as A001405.

Cf. A056594, A138350, A210628, A211278.

Cf. A000108, A091867, A146559, A126930.

Programs

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 *)

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)};

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)