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A000320 Generalized tangent numbers d(5,n).

%I A000320 M3722 N1521 #34 Nov 22 2021 04:03:25
%S A000320 4,272,55744,23750912,17328937984,19313964388352,30527905292468224,
%T A000320 64955605537174126592,179013508069217017790464,
%U A000320 620314831396713435870789632,2639743384489464189324523208704,13533573366345611477262311433961472,82274260343572247169162187576069586944
%N A000320 Generalized tangent numbers d(5,n).
%D A000320 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000320 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000320 Lars Blomberg, <a href="/A000320/b000320.txt">Table of n, a(n) for n = 1..189</a>
%H A000320 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223295-5">Generalized Euler and class numbers</a>. Math. Comp. 21 (1967)  689-694.
%H A000320 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1968-0227093-9">Corrigenda to: "Generalized Euler and class numbers"</a>, Math. Comp. 22 (1968), 699
%H A000320 D. Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
%F A000320 a(n) = (2*n-1)!*[x^(2*n-1)](sec(5*x)*(sin(x) + sin(3*x))). - _Peter Luschny_, Nov 21 2021
%p A000320 egf := sec(5*x)*(sin(x) + sin(3*x)): ser := series(egf, x, 26):
%p A000320 seq((2*n-1)!*coeff(ser, x, 2*n-1), n = 1..13); # _Peter Luschny_, Nov 21 2021
%t A000320 nmax = 15; km0 = 10; Clear[dd]; L[a_, s_, km_] := Sum[JacobiSymbol[-a, 2 k + 1]/(2k+1)^s, {k, 0, km}]; d[a_ /; a>1, n_, km_] := (2n-1)! L[-a, 2n, km] (2a/Pi)^(2n)/Sqrt[a] // Round; dd[km_] := dd[km] = Table[d[5, n, km], {n, 1, nmax}]; dd[km0]; dd[km = 2km0]; While[dd[km] != dd[km/2, km = 2 km]]; A000320 = dd[km] (* _Jean-François Alcover_, Feb 07 2016 *)
%Y A000320 Cf. A000318, A000187, A349265, A349264.
%K A000320 nonn
%O A000320 1,1
%A A000320 _N. J. A. Sloane_
%E A000320 Formula producing A000326, rather than this sequence, deleted by _Sean A. Irvine_, Sep 09 2010
%E A000320 a(10)-a(13) from _Lars Blomberg_, Sep 07 2015