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URL: https://oeis.org/A240586

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A240586
Expansion of (((8-8 / sqrt(1-8*x)) / (2*sqrt(8*x+2*sqrt(1-8*x)+2))+4 / sqrt(1-8*x))*((x*(sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1))-4*x^2)) / (sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1)^2.
1
1, 4, 22, 140, 950, 6692, 48284, 354216, 2630310, 19713188, 148817524, 1130011896, 8621650492, 66043991080, 507628779896, 3913088587472, 30240258982662, 234210742764964, 1817484391184900, 14128074297880536, 109992814064010196, 857525947713607096, 6693820044841440008
OFFSET
1,2
LINKS
FORMULA
a(n) = Sum_{m=1..n} binomial(n-1, m-1)*Sum_{j=0..m} j*(-1)^(j-m)*binomial(m, j)*Sum_{k=1..n} binomial(k, n-k) * binomial(2*k+j-1, k+j-1) / (k+j).
A(x) = B'(x) * (x*B(x)-x^2) / B(x)^2, where B(x) = (-1-sqrt(1-8*x)+sqrt(2+2*sqrt(1-8*x)+8*x))/4, B(x)/x is g.f. of A186997.
a(n) ~ 8^(n-1) * (sqrt(3)-1) / sqrt(Pi*n). - Vaclav Kotesovec, Apr 12 2014
MATHEMATICA
Rest[CoefficientList[Series[(((8-8 / Sqrt[1-8*x]) / (2*Sqrt[8*x+2*Sqrt[1-8*x]+2])+4 / Sqrt[1-8*x])*((x*(Sqrt[8*x+2*Sqrt[1-8*x]+2]-Sqrt[1-8*x]-1))-4*x^2)) / (Sqrt[8*x+2*Sqrt[1-8*x]+2]-Sqrt[1-8*x]-1)^2, {x, 0, 20}], x]] (* Vaclav Kotesovec, Apr 12 2014 *)
PROG
(Maxima)
a(n):=sum((sum(j*(sum((binomial(k, n-k)*binomial(2*k+j-1, k+j-1)) / (k+j), k, 1, n))*(-1)^(j-m)*binomial(m, j), j, 0, m))*binomial(n-1, m-1), m, 1, n);
(PARI) my(x='x+O('x^50)); Vec((((8-8 / sqrt(1-8*x)) / (2*sqrt(8*x+2*sqrt(1-8*x)+2))+4 / sqrt(1-8*x))*((x*(sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1))-4*x^2)) / (sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1)^2) \\ G. C. Greubel, Apr 05 2017
CROSSREFS
Sequence in context: A187254 A325453 A216712 * A369156 A002293 A181784
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Apr 08 2014
STATUS
approved