OFFSET
0,3
COMMENTS
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)!!.
This sequence is the linking pin between the a(n) formulas of the ED1, ED2, ED3 and ED4 array rows, see A167552, A167565, A167580 and A167591. - Johannes W. Meijer, Nov 23 2009
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
P. S. Bruckman, An interesting sequence of numbers derived from various generating functions, Fib. Quart., 10 (1972), 169-181.
R. J. Mathar, Numerical Representation of the Incomplete Gamma Function of Complex Argument, arXiv:math/0306184 [math.NA], 2003-2004; cf. Eq. 22.
FORMULA
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
E.g.f.: 1/(sqrt(1+2*x)*(1-2*x)). - Vladeta Jovovic, Oct 12 2003
a(n) = (2^n)*n!*A123746(n)/A046161(n) = (2^n)*n!*Sum_{k=0..n} binomial(2*k,k)*(-1/4)^k. From the e.g.f. - Wolfdieter Lang, Oct 06 2008
a(n) = A091520(n) * n! / 2^n. - Michael Somos, Mar 17 2011
EXAMPLE
arctan(sqrt(2*x/(1-x)))/sqrt(2*x*(1-x)) = 1 + 1/3*x + 7/15*x^2 + 9/35*x^3 + ...
MAPLE
# 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
MATHEMATICA
a[n_] := a[n] = (-1)^n*(2n - 1)!! + 2n*a[n - 1]; a[0] = 1; Table[ a[n], {n, 0, 14}] (* Jean-François Alcover, Dec 01 2011, after R. J. Mathar *)
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 *)
PROG
(Haskell)
a003148 n = a003148_list !! n
a003148_list = 1 : 1 : zipWith (+) (tail a003148_list)
(zipWith (*) (tail a002943_list) a003148_list)
-- Reinhard Zumkeller, Nov 22 2011
(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
[a(n) for n in range(31)] # G. C. Greubel, Nov 04 2022
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
a(16)-a(20) from Andrew Howroyd, Feb 05 2018
STATUS
approved