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%I M5027 N2169 #34 Nov 22 2021 04:03:34
%S 1,16,1280,249856,90767360,52975108096,45344872202240,
%T 53515555843342336,83285910482761809920,165262072909347030040576,
%U 407227428060372417275494400,1219998300294918683087199010816,4366953142363907901751614431559680,18406538229888710811704852978971181056
%N Generalized Euler numbers c(4,n).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H 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 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 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 a(n) = A000364(n)*16^n. - _Philippe Deléham_, Oct 27 2006
%F a(n) = (2*n)!*[x^(2*n)](sec(4*x)). - _Peter Luschny_, Nov 21 2021
%p egf := sec(4*x): ser := series(egf, x, 26):
%p seq((2*n)!*coeff(ser, x, 2*n), n = 0..11); # _Peter Luschny_, Nov 21 2021
%t a0 = 4; nmax = 20; km0 = nmax; Clear[cc]; L[a_, s_, km_] := Sum[ JacobiSymbol[-a, 2*k+1]/(2*k+1)^s, {k, 0, km}]; c[a_, n_, km_] := 2^(2*n +1)*Pi^(-(2*n)-1)*(2*n)!*a^(2*n+1/2)*L[a, 2*n+1, km] // Round; cc[km_] := cc[km] = Table[c[a0, n, km], {n, 0, nmax}]; cc[km0]; cc[km = 2 km0]; While[cc[km] != cc[km/2, km = 2 km]]; A000490 = cc[km] (* _Jean-François Alcover_, Feb 05 2016 *)
%Y Cf. A000187, A000436, A000318, A349264.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 02 2000
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