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A002085 From the expansion of sinh(x) / cos(x): a(n) = odd part of A002084(n).

%I A002085 M4620 N1973 #37 Jul 22 2025 10:31:50
%S A002085 1,1,9,39,1141,12721,804309,17113719,1886573641,65373260641,
%T A002085 11127809595009,570506317184199,138730500808639741,
%U A002085 9867549661639871761,3247323803322747092109,305991103023048833011479,130958280255560469436519441,15789929076277491342686566081
%N A002085 From the expansion of sinh(x) / cos(x): a(n) = odd part of A002084(n).
%D A002085 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002085 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002085 T. D. Noe, <a href="/A002085/b002085.txt">Table of n, a(n) for n = 0..50</a>
%H A002085 Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, <a href="https://www.wisdom.weizmann.ac.il/~fraenkel/Papers/WythoffWisdomJune62016.pdf">Wythoff Wisdom</a>, 43 pages, no date, unpublished.
%H A002085 Eric Duchene, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, and Urban Larsson, <a href="/A001950/a001950.pdf">Wythoff Wisdom</a>, unpublished, no date. [Cached copy, with permission]
%H A002085 J. M. Gandhi, <a href="https://dx.doi.org/10.4153/CMB-1970-059-9">The coefficients of sinh x/ cos x</a>. Canad. Math. Bull. 13 1970 305-310.
%F A002085 Divide A002084(n) by 4^(floor((n+1)/4)).
%t A002085 a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sinh[x] / Cos[x], {x, 0, 2 n + 1}] (2 n + 1)! / 2^(n + Mod[n, 2])] (* _Michael Somos_, Apr 10 2011 *)
%o A002085 (PARI) {a(n) = local(A, m); if( n<0, 0, m = 2*n + 1; A = x * O(x^m) ; 2^(n + 1 - n%2) * m! * polcoeff( sinh( x/2 + A) / cos( x/2 + A), m))} /* _Michael Somos_, Apr 10 2011 */
%Y A002085 Cf. A002084.
%K A002085 nonn,easy
%O A002085 0,3
%A A002085 _N. J. A. Sloane_
%E A002085 Added more terms from b-file. - _David Radcliffe_, Jul 21 2025