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A210736 Expansion of (1 + sqrt( (1 + 2*x) / (1 - 2*x))) / 2 in powers of x. 8
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 (list; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A056202 A001405 A126930 * A036557 A173125 A047131
KEYWORD
nonn
AUTHOR
Michael Somos, May 10 2012
STATUS
approved

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Last modified July 23 13:35 EDT 2024. Contains 374549 sequences. (Running on oeis4.)