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A110491
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Expansion of e.g.f.: sqrt(1+2x)/sqrt(1-2x).
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1
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1, 2, 4, 24, 144, 1440, 14400, 201600, 2822400, 50803200, 914457600, 20118067200, 442597478400, 11507534438400, 299195895398400, 8975876861952000, 269276305858560000, 9155394399191040000, 311283409572495360000
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OFFSET
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0,2
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COMMENTS
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Row sums of exponential Riordan array [1, arctanh(2x)]. - Paul Barry, Apr 17 2008
Conjecture: {a(n-1), n>=1} is the T-transform of A093178, where T maps a sequence {b(n), n>=1} to the sequence {c(n)} defined by c(n) = det(M_n), where M_n is the n X n matrix with elements M_n(i,j) = b(2*j) for i>j and M_n(i,j) = b(i+j-1) for i<=j. - Lechoslaw Ratajczak, Aug 04 2021
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LINKS
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FORMULA
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E.g.f.: sqrt((1+2x)/(1-2x)); a(n)=2*n!*binomial(n-1, floor((n-1)/2))+0^n.
The sequence 0,1,0,2,0,4,... has e.g.f. arctanh(x). - Paul Barry, Apr 17 2008
D-finite with recurrence a(n) -2*a(n-1) -4*(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Sep 20 2012
E.g.f. satisfies 2*g(x)+(4*x^2-1)*g'(x) = 0, from which Mathar's recurrence follows. (End)
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MAPLE
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S:= series(sqrt(1+2*x)/sqrt(1-2*x), x, 31):
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MATHEMATICA
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With[{nn=20}, CoefficientList[Series[Sqrt[1+2x]/Sqrt[1-2x], {x, 0, nn}], x] Range[0, nn]!] (* or *) Join[{1}, Table[2n!Binomial[n-1, Floor[(n-1)/2]], {n, 20}]] (* Harvey P. Dale, Nov 11 2011 *)
Table[2^n Binomial[1/2, n] n! Hypergeometric2F1[1/2, -n, 3/2 - n, -1], {n, 0, 20}] (* Benedict W. J. Irwin, Jun 06 2016 *)
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PROG
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(PARI) my(x='x+O('x^25)); Vec(serlaplace(sqrt(1+2*x)/sqrt(1-2*x))) \\ Michel Marcus, Aug 05 2021
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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