A058962 a(n) = 2^(2*n)*(2*n+1).

1, 12, 80, 448, 2304, 11264, 53248, 245760, 1114112, 4980736, 22020096, 96468992, 419430400, 1811939328, 7784628224, 33285996544, 141733920768, 601295421440, 2542620639232, 10720238370816, 45079976738816, 189115999977472, 791648371998720

Offset: 0

Author

N. J. A. Sloane, Jan 13 2001

Keywords

• nonn
• easy

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).

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

Programs

Magma

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

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 *)

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 */

Formula

Central terms of the triangle in A118416: a(n) = A118416(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