0,3

Numerators of sequence of fractions with e.g.f. 1/((1-x)*(1+x)^(1/2)). The denominators are suuccessive 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)!!.

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.yu), Oct 12 2003

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: A118101 A034536 A035081 this_sequence A033910 A015817 A103253

Adjacent sequences: A003145 A003146 A003147 this_sequence A003149 A003150 A003151

nonn,nice,easy

njas