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A116406 Expansion of ((1+x-2x^2)+(1+x)sqrt(1-4x^2))/(2(1-4x^2)). 2
1, 1, 2, 3, 7, 11, 26, 42, 99, 163, 382, 638, 1486, 2510, 5812, 9908, 22819, 39203, 89846, 155382, 354522, 616666, 1401292, 2449868, 5546382, 9740686, 21977516, 38754732, 87167164, 154276028, 345994216, 614429672, 1374282019, 2448023843 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Interleaving of A114121 and A032443. Row sums of A116405. Binomial transform is A116409.

Appears to be the number of n-digit binary numbers not having more zeros than ones; equivalently, the number of unrestricted Dyck paths of length n not going below the axis. - Ralf Stephan, Mar 25 2008

LINKS

Table of n, a(n) for n=0..33.

FORMULA

a(n)=A114121(n/2)*(1+(-1)^n)/2+A032443((n-1)/2)*(1-(-1)^n)/2.

a(n)=sum{k=0..floor(n/2), binomial(n-1,k)}; - Paul Barry, Oct 06 2007

Conjecture: n*(n-3)*a(n) +2*(-n^2+4*n-2)*a(n-1) -4*(n-2)^2*a(n-2) +8*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 28 2014

a(n) ~ 2^(n-2) * (1 + (3+(-1)^n)/sqrt(2*Pi*n)). - Vaclav Kotesovec, May 30 2016

MATHEMATICA

CoefficientList[Series[((1+x-2x^2)+(1+x)Sqrt[1-4x^2])/(2(1-4x^2)), {x, 0, 40}], x] (* Harvey P. Dale, Aug 16 2012 *)

CROSSREFS

Sequence in context: A101173 A294451 A005246 * A112843 A036651 A049454

Adjacent sequences:  A116403 A116404 A116405 * A116407 A116408 A116409

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 13 2006

STATUS

approved

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Last modified January 29 10:34 EST 2020. Contains 331337 sequences. (Running on oeis4.)