This site is supported by donations to The OEIS Foundation. Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A003148 a(n+1) = a(n) + 2n*(2n+1)*a(n-1), with a(0) = a(1) = 1. (Formerly M4389) 13

%I M4389

%S 1,1,7,27,321,2265,37575,390915,8281665,114610545,2946939975,

%T 51083368875,1542234996225,32192256321225,1114841223671175,

%U 27254953356505875,1064057291370698625,29845288035840902625,1296073464766972266375,41049997128507054562875

%N a(n+1) = a(n) + 2n*(2n+1)*a(n-1), with a(0) = a(1) = 1.

%C Numerators of sequence of fractions with e.g.f. 1/((1-x)*(1+x)^(1/2)). The denominators are successive powers of 2.

%C 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)!!.

%C 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

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

%H T. D. Noe, <a href="/A003148/b003148.txt">Table of n, a(n) for n=0..100</a>

%H P. S. Bruckman, <a href="http://www.fq.math.ca/Scanned/10-2/bruckman.pdf">An interesting sequence of numbers derived from various generating functions</a>, Fib. Quart., 10 (1972), 169-181.

%H R. J. Mathar, <a href="https://arxiv.org/abs/math/0306184">Numerical Representation of the Incomplete Gamma Function of Complex Argument</a>, arXiv:math/0306184 [math.NA], 2003-2004; cf. Eq. 22.

%F a(n) = (-1)^n*(2n-1)!! + 2n*a(n-1) with (2n-1)!! = 1*3*5*..*(2n-1) the double factorial. - _R. J. Mathar_, Jun 12 2003

%F a(n) = [(2n+1)!!/4] Int ([cos(phi)]^n cos(phi/2), phi=-Pi..Pi). - _R. J. Mathar_, Jun 30 2003

%F a(n) = (2n+1)!! 2F1(-n, 1/2;3/2;2). - _R. J. Mathar_, Jun 30 2003

%F 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_, Jun 30 2003

%F E.g.f.: 1/(sqrt(1+2*x)*(1-2*x)). - _Vladeta Jovovic_, Oct 12 2003

%F 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. - _Wolfdieter Lang_, Oct 06 2008.

%F a(n) = A091520(n) * n! / 2^n. - _Michael Somos_, Mar 17 2011

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

%p # 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

%t a[n_] := a[n] = (-1)^n*(2n - 1)!! + 2n*a[n - 1]; a = 1; Table[ a[n], {n, 0, 14}] (* _Jean-François Alcover_, Dec 01 2011, after _R. J. Mathar_ *)

%t a[ n_] := If[ n < 0, 0, (2 n + 1)!! Hypergeometric2F1[ -n, 1/2, 3/2, 2]]; (* _Michael Somos_, Apr 20 2018 *)

%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / ((1 - 2 x) Sqrt[1 + 2 x]), {x, 0, n}]]; (* _Michael Somos_, Apr 20 2018 *)

%t RecurrenceTable[{a==a==1,a[n+1]==a[n]+2n(2n+1)a[n-1]},a,{n,20}] (* _Harvey P. Dale_, Jul 27 2019 *)

%o a003148 n = a003148_list !! n

%o a003148_list = 1 : 1 : zipWith (+) (tail a003148_list)

%o (zipWith (*) (tail a002943_list) a003148_list)

%o -- _Reinhard Zumkeller_, Nov 22 2011

%o (PARI) Vec(serlaplace(1/(sqrt(1+2*x + O(x^20))*(1-2*x)))) \\ _Andrew Howroyd_, Feb 05 2018

%Y Contribution from _Johannes W. Meijer_, Nov 23 2009: (Start)

%Y Appears in A167552, A167565, A167580 and A167591.

%Y Equals A049606*A123746.

%Y (End)

%Y Cf. A002943.

%K nonn,nice,easy

%O 0,3

%A _N. J. A. Sloane_

%E a(16)-a(20) from _Andrew Howroyd_, Feb 05 2018

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 17:42 EST 2019. Contains 329768 sequences. (Running on oeis4.)