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A119967
A transform of the central binomial coefficients C(n,floor(n/2)).
1
1, 1, 2, 4, 9, 21, 49, 113, 259, 595, 1376, 3202, 7479, 17499, 40986, 96116, 225755, 531115, 1251310, 2951556, 6968883, 16468775, 38951925, 92204241, 218426037, 517799861, 1228280392, 2915346934, 6923469409, 16450694861, 39107365561
OFFSET
0,3
COMMENTS
Inverse binomial transform is A001405 with interpolated zeros.
Hankel transform is 1,1,1,-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,... [Paul Barry, Feb 21 2009]
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
FORMULA
G.f.: 2*(1-x)/(1-2*x-x^2+sqrt(1-4*x+6*x^2-4*x^3-3*x^4)).
a(n) = sum{k=0..floor(n/2), C(n,2k)*C(k,floor(k/2))}.
G.f.: 1/(1-x-x^2/(1-x-x^2/(1-x+x^2/(1-x+x^2/(1-x-x^2/(1-x-x^2/(1-x+x^2/(1-x+x^2/(1-... (continued fraction). [Paul Barry, Feb 21 2009]
Conjecture: (n+2)*(n-3)*a(n) +(2+2*n-n^2)*a(n-1) +(3*n-5)*(n-2)*a(n-2) -(n-2)*(n-3)*a(n-3) -(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 14 2011
a(n) ~ sqrt(5+7/sqrt(2)) * (1+sqrt(2))^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 01 2014
MATHEMATICA
CoefficientList[Series[2*(1-x)/(1-2*x-x^2+Sqrt[1-4*x+6*x^2-4*x^3-3*x^4]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)
PROG
(PARI) x='x+O('x^50); Vec(2*(1-x)/(1-2*x-x^2 + sqrt(1-4*x+6*x^2-4*x^3-3*x^4))) \\ G. C. Greubel, Mar 19 2017
CROSSREFS
Sequence in context: A182904 A281425 A101891 * A266232 A052921 A219150
KEYWORD
nonn
AUTHOR
Paul Barry, May 31 2006
STATUS
approved