login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A058962 a(n) = 2^(2*n)*(2*n+1). 17
1, 12, 80, 448, 2304, 11264, 53248, 245760, 1114112, 4980736, 22020096, 96468992, 419430400, 1811939328, 7784628224, 33285996544, 141733920768, 601295421440, 2542620639232, 10720238370816, 45079976738816, 189115999977472, 791648371998720 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Denominators in expansion of -1/2*i*Pi+i*arcsin((1+1/4*x^2)/(1-1/4*x^2)), where i=sqrt(-1); numerators are all 1.

Bisection of A001787. That is, a(n) = A001787(2n+1). - Graeme McRae, Jul 12 2006

Denominators of odd terms in expansion of 2*arctanh(s/2); numerators are all 1. - Gerry Martens, Jul 26 2015

Reciprocals of coefficients of Taylor series expansion of sinh(x/2) / (x/2). - Tom Copeland, Feb 03 2016

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..200

G. A. Campbell, Physical theory of the electric wave-filter, Bell Syst. Tech. J., 1 (1922), 1-32, see Eq. (15d). Also reprinted in M. E. Van Valkebburg, ed., Circuit Theory, Dowden, Hutchinson and Ross, 1974.

Index entries for linear recurrences with constant coefficients, signature (8,-16).

FORMULA

Central terms of the triangle in A118413: a(n) = A118413(2*n+1,n+1). - Reinhard Zumkeller, Apr 27 2006

Sum_{n>=0} 1/a(n) = log(3). - Jaume Oliver Lafont, May 22 2007; corrected by Jaume Oliver Lafont, Jan 26 2009

a(n) = 4((2n+1)/(2n-1))*a(n-1) = 4*a(n-1)+2^(2n+1) = 8*a(n-1)-16*a(n-2). - Jaume Oliver Lafont, Dec 09 2008

G.f.: (1+4*x)/(1-4*x)^2. - Jaume Oliver Lafont, Jan 29 2009

E.g.f.: exp(4*x)*(1+8*x). - Robert Israel, Aug 10 2015

a(n) = -a(-1-n) * 4^(2*n+1) for all n in Z. - Michael Somos, Jun 18 2017

a(n) = Sum_{k = 0..n} (2*k + 1)^2*binomial(2*n + 1, n - k). - Peter Bala, Feb 25 2019

Sum_{n>=0} (-1)^n/a(n) = 2 * arctan(1/2) = 2 * A073000. - Amiram Eldar, Jul 03 2020

MATHEMATICA

a[n_] := 1/SeriesCoefficient[2 ArcTanh[s/2], {s, 0, n}]

Table[a[n], {n, 1, 40, 2}] (* Gerry Martens, Jul 26 2015 *)

Table[2^(2 n) (2 n + 1), {n, 0, 40}] (* Vincenzo Librandi, Aug 08 2015 *)

a[ n_] := With[{m = 2 n + 2}, If[ n < 0, -a[-1 - n] 4^(m - 1), m! SeriesCoefficient[ x^2 D[x Sinc[I x]^2, x]/2, {x, 0, m}]]]; (* Michael Somos, Jun 18 2017 *)

PROG

(PARI) { for (n = 0, 200, write("b058962.txt", n, " ", 2^(2*n)*(2*n+1)); ) } \\ Harry J. Smith, Jun 24 2009

(PARI) first(m)=vector(m, n, n--; 2^(2*n)*(2*n+1)) /* Anders Hellström, Aug 10 2015 */

(Magma) [2^(2*n)*(2*n+1) : n in [0..30]]; // Wesley Ivan Hurt, Aug 07 2015

(PARI) A058962(n)=2^(2*n)*(2*n+1) \\ M. F. Hasler, Aug 11 2015

(PARI) {a(n) = my(m = 2*n + 2); if( n<0, -a(-1 - n) * 4^(m - 1), m! * polcoeff( x^2 * deriv(x * sinc(I*x + x * O(x^m))^2, x) / 2, m))}; /* Michael Somos, Jun 18 2017 */

CROSSREFS

Cf. A001787, A002391, A073000, A118413, A118415.

Cf. A154920. - Jaume Oliver Lafont, Jan 29 2009

Factor of the LS1[-2,n] matrix coefficients in A160487. - Johannes W. Meijer, May 24 2009

Sequence in context: A160559 A038734 A258591 * A203486 A187011 A277783

Adjacent sequences: A058959 A058960 A058961 * A058963 A058964 A058965

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Jan 13 2001

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 09:20 EST 2022. Contains 358607 sequences. (Running on oeis4.)