0,3

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)!!.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. S. Bruckman, An interesting sequence of numbers derived from various generating functions, Fib. Quart., 10 (1972), 169-181.

T. D. Noe, Table of n, a(n) for n=0..100

R. J. Mathar, Numerical Representation of the Incomplete Gamma Function of Complex Argument, cf. Eq. 22.

a(n) = (-1)^n (2n-1)!! + 2na(n-1) with (2n-1)!! = 1*3*5*..*(2n-1) the double factorial. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 12 2003

a(n)=[(2n+1)!!/4] Int ([cos(phi)]^n cos(phi/2), phi=-Pi..Pi). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 30 2003

a(n) = (2n+1)!! 2F1(-n, 1/2;3/2;2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 30 2003

In terms of the (terminating) Gauss hypergeometric function/series 2F1(., .; ., .) a(n) is a special case of the family of integer sequences defined by a(m, n) = [(2n+2m+1)!!/(2m+1)] 2F1(-n, m+1/2; m+3/2; 2), m=0, 1, 2, ..., n=0, 1, 2, ...; a(n) = a(0, n); a(m, n) = [(2n+2m+1)!!/4] Int ([sin(phi/2)]^(2m) [cos(phi)]^n cos(phi/2), phi=-Pi. .Pi); 4(n+1)a(m, n) = (2m-1) a(m-1, n+1)+(-1)^n (2n+2m+1)!!. a(0, n) = this sequence, a(1, n) = A077568, a(2, n) = A084543. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 30 2003

E.g.f.: 1/(sqrt(1+2*x)*(1-2*x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 12 2003

a(n)=(2^n)*n!*A123746(n)/A046161(n) = (2^n)*n!*sum(binomial(2*k,k)*(-1/4)^k,k=0..n). From the e.g.f. - W. Lang, Oct 06 2008.

arctan(sqrt(2*x/(1-x)))/sqrt(2*x*(1-x)) = 1 + 1/3*x + 7/15*x^2 + 9/35*x^3 + ...

# 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

Sequence in context: A034536 A151496 A035081 this_sequence A033910 A152578 A155712

Adjacent sequences: A003145 A003146 A003147 this_sequence A003149 A003150 A003151

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).