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%I #73 Nov 07 2023 13:36:06
%S 1,12,80,448,2304,11264,53248,245760,1114112,4980736,22020096,
%T 96468992,419430400,1811939328,7784628224,33285996544,141733920768,
%U 601295421440,2542620639232,10720238370816,45079976738816,189115999977472,791648371998720
%N a(n) = 2^(2*n)*(2*n+1).
%C 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.
%C Bisection of A001787. That is, a(n) = A001787(2n+1). - _Graeme McRae_, Jul 12 2006
%C Denominators of odd terms in expansion of 2*arctanh(s/2); numerators are all 1. - _Gerry Martens_, Jul 26 2015
%C Reciprocals of coefficients of Taylor series expansion of sinh(x/2) / (x/2). - _Tom Copeland_, Feb 03 2016
%H Harry J. Smith, <a href="/A058962/b058962.txt">Table of n, a(n) for n = 0..200</a>
%H G. A. Campbell, <a href="https://archive.org/details/bstj1-2-1">Physical theory of the electric wave-filter</a>, 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.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-16).
%F Central terms of the triangle in A118416: a(n) = A118416(2*n+1, n+1) - _Reinhard Zumkeller_, Apr 27 2006
%F Sum_{n>=0} 1/a(n) = log(3). - _Jaume Oliver Lafont_, May 22 2007; corrected by _Jaume Oliver Lafont_, Jan 26 2009
%F 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
%F G.f.: (1+4*x)/(1-4*x)^2. - _Jaume Oliver Lafont_, Jan 29 2009
%F E.g.f.: exp(4*x)*(1+8*x). - _Robert Israel_, Aug 10 2015
%F a(n) = -a(-1-n) * 4^(2*n+1) for all n in Z. - _Michael Somos_, Jun 18 2017
%F a(n) = Sum_{k = 0..n} (2*k + 1)^2*binomial(2*n + 1, n - k). - _Peter Bala_, Feb 25 2019
%F Sum_{n>=0} (-1)^n/a(n) = 2 * arctan(1/2) = 2 * A073000. - _Amiram Eldar_, Jul 03 2020
%t a[n_] := 1/SeriesCoefficient[2 ArcTanh[s/2],{s,0,n}]
%t Table[a[n], {n, 1, 40, 2}] (* _Gerry Martens_, Jul 26 2015 *)
%t Table[2^(2 n) (2 n + 1), {n, 0, 40}] (* _Vincenzo Librandi_, Aug 08 2015 *)
%t 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 *)
%o (PARI) { for (n = 0, 200, write("b058962.txt", n, " ", 2^(2*n)*(2*n+1)); ) } \\ _Harry J. Smith_, Jun 24 2009
%o (PARI) first(m)=vector(m, n, n--; 2^(2*n)*(2*n+1)) /* _Anders Hellström_, Aug 10 2015 */
%o (Magma) [2^(2*n)*(2*n+1) : n in [0..30]]; // _Wesley Ivan Hurt_, Aug 07 2015
%o (PARI) A058962(n)=2^(2*n)*(2*n+1) \\ _M. F. Hasler_, Aug 11 2015
%o (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 */
%Y Cf. A001787, A002391, A073000, A118413, A118415.
%Y Cf. A154920. - _Jaume Oliver Lafont_, Jan 29 2009
%Y Factor of the LS1[-2,n] matrix coefficients in A160487. - _Johannes W. Meijer_, May 24 2009
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 13 2001
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