|
| |
|
|
A003148
|
|
a(n+1) = a(n) + 2n(2n+1)a(n-1).
(Formerly M4389)
|
|
13
| |
|
|
1, 1, 7, 27, 321, 2265, 37575, 390915, 8281665, 114610545, 2946939975, 51083368875, 1542234996225, 32192256321225, 1114841223671175
(list; graph; refs; listen; history; internal format)
|
|
|
|
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)!!.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 23 2009: (Start)
This sequence is the linking pin between the a(n) formulae of the ED1, ED2, ED3 and ED4 array rows, see A167552, A167565, A167580 and A167591.
(End)
|
|
|
REFERENCES
| P. S. Bruckman, An interesting sequence of numbers derived from various generating functions, Fib. Quart., 10 (1972), 169-181.
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
R. J. Mathar, Numerical Representation of the Incomplete Gamma Function of Complex Argument, cf. Eq. 22.
|
|
|
FORMULA
| 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.
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}] (* From Jean-François Alcover, Dec 01 2011, after R. J. Mathar *)
|
|
|
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
|
|
|
CROSSREFS
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 23 2009: (Start)
Appears in A167552, A167565, A167580 and A167591.
Equals A049606*A123746.
(End)
Cf. A002943.
Sequence in context: A193257 A173193 A196323 * A033910 A196647 A152578
Adjacent sequences: A003145 A003146 A003147 * A003149 A003150 A003151
|
|
|
KEYWORD
| nonn,nice,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
