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A003148
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a(n+1) = a(n) + 2n*(2n+1)*a(n-1), with a(0) = a(1) = 1.
(Formerly M4389)
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13
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1, 1, 7, 27, 321, 2265, 37575, 390915, 8281665, 114610545, 2946939975, 51083368875, 1542234996225, 32192256321225, 1114841223671175, 27254953356505875, 1064057291370698625, 29845288035840902625, 1296073464766972266375, 41049997128507054562875
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OFFSET
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0,3
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COMMENTS
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Numerators of sequence of fractions with e.g.f. 1/((1-x)*(1+x)^(1/2)). The denominators are successive powers of 2.
a(n) is the coefficient of x^n in arctan(sqrt(2*x/(1-x)))/sqrt(2*x*(1-x)) multiplied by (2*n+1)!!.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = (-1)^n*(2n-1)!! + 2*n*a(n-1) with (2n-1)!! = 1*3*5*..*(2n-1) the double factorial. - R. J. Mathar, Jun 12 2003
a(n) = ((2*n+1)!!/4) * Integral_{-Pi..Pi} cos(x)^n * cos(x/2) dx. - R. J. Mathar, Jun 30 2003
a(n) = (2n+1)!! 2F1(-n, 1/2;3/2;2). - R. J. Mathar, Jun 30 2003
In terms of the (terminating) Gauss hypergeometric function/series, 2F1(., .; .; 2), a(n) is a special case of the family of integer sequences defined by a(m, n) = ((2*n+2*m+1)!!/(2*m+1)) * 2F1(-n, m+1/2; m+3/2; 2), for m >= 0, n >= 0. An integral form can be seen as a(m, n) = ((2*n+2*m+1)!!/4) * Integral_{-Pi..Pi} ((sin(x/2))^(2*m) * (cos(x))^n * cos(x/2) dx. A recurrence property is 4*(n+1)*a(m, n) = (2*m-1)*a(m-1, n+1) + (-1)^n*(2*n+2*m+1)!!. Sequences that have these properties are a(0, n) = this sequence, a(1, n) = A077568, a(2, n) = A084543. - R. J. Mathar, Jun 30 2003
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EXAMPLE
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arctan(sqrt(2*x/(1-x)))/sqrt(2*x*(1-x)) = 1 + 1/3*x + 7/15*x^2 + 9/35*x^3 + ...
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MAPLE
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# double factorial of odd "l" df := proc(l) local n; n := iquo(l, 2); RETURN( factorial(l)/2^n/factorial(n)); end: x := 1; for n from 1 to 15 do if n mod 2 = 0 then x := 2*n*x+df(2*n-1); else x := 2*n*x-df(2*n-1); fi; print(x); od; quit
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, (2 n + 1)!! Hypergeometric2F1[ -n, 1/2, 3/2, 2]]; (* Michael Somos, Apr 20 2018 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / ((1 - 2 x) Sqrt[1 + 2 x]), {x, 0, n}]]; (* Michael Somos, Apr 20 2018 *)
RecurrenceTable[{a[0]==a[1]==1, a[n+1]==a[n]+2n(2n+1)a[n-1]}, a, {n, 20}] (* Harvey P. Dale, Jul 27 2019 *)
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PROG
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(Haskell)
a003148 n = a003148_list !! n
a003148_list = 1 : 1 : zipWith (+) (tail a003148_list)
(zipWith (*) (tail a002943_list) a003148_list)
(PARI) Vec(serlaplace(1/(sqrt(1+2*x + O(x^20))*(1-2*x)))) \\ Andrew Howroyd, Feb 05 2018
(Magma) [n le 2 select 1 else Self(n-1) + 2*(n-2)*(2*n-3)*Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 04 2022
(SageMath)
@CachedFunction
def a(n): return 1 if (n<2) else a(n-1) + 2*(n-1)*(2*n-1)*a(n-2) # a = A003148
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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