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%I M5082 N2199 #49 Oct 14 2023 21:11:02
%S 1,1,19,1513,315523,136085041,105261234643,132705221399353,
%T 254604707462013571,705927677520644167681,2716778010767155313771539,
%U 14050650308943101316593590153,95096065132610734223282520762883,823813936407337360148622860507620561
%N Generalized Euler numbers.
%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 Alois P. Heinz, <a href="/A002115/b002115.txt">Table of n, a(n) for n = 0..166</a>
%H Takao Komatsu and Ram Krishna Pandey, <a href="https://doi.org/10.3934/math.2021390">On hypergeometric Cauchy numbers of higher grade</a>, AIMS Mathematics (2021) Vol. 6, Issue 7, 6630-6646.
%H D. H. Lehmer, <a href="https://www.jstor.org/stable/1968647">Lacunary recurrence formulas for the numbers of Bernoulli and Euler</a>, Annals Math., 36 (1935), 637-649.
%F E.g.f.: Sum_{n >= 0} a(n)*x^n/(3*n)! = 1/((1/3)*exp(-x^(1/3)) + (2/3)*exp((1/2)*x^(1/3))*cos((1/2)*3^(1/2)*x^(1/3))). - _Vladeta Jovovic_, Feb 13 2005
%F E.g.f.: 1/U(0) where U(k) = 1 - x/(6*(6*k+1)*(3*k+1)*(2*k+1) - 6*x*(6*k+1)*(3*k+1)*(2*k+1)/(x - 12*(6*k+5)*(3*k+2)*(k+1)/U(k+1))); (continued fraction, 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Oct 04 2012
%F Alternating row sums of A278073. - _Peter Luschny_, Sep 07 2017
%F a(n) = A178963(3n). - _Alois P. Heinz_, Aug 12 2019
%F a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(3*n,3*k) * a(n-k). - _Ilya Gutkovskiy_, Jan 27 2020
%F a(n) = (3*n)! * [x^(3*n)] hypergeom([], [1/3, 2/3], (-x/3)^3)^(-1). - _Peter Luschny_, Mar 13 2023
%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1, `if`(t=0,
%p add(b(u-j, o+j-1, irem(t+1, 3)), j=1..u),
%p add(b(u+j-1, o-j, irem(t+1, 3)), j=1..o)))
%p end:
%p a:= n-> b(3*n, 0$2):
%p seq(a(n), n=0..17); # _Alois P. Heinz_, Aug 12 2019
%p # Alternative:
%p h := 1 / hypergeom([], [1/3, 2/3], (-x/3)^3): ser := series(h, x, 40):
%p seq((3*n)! * coeff(ser, x, 3*n), n = 0..13); # _Peter Luschny_, Mar 13 2023
%t max = 12; f[x_] := 1/(1/3*Exp[-x^(1/3)] + 2/3*Exp[1/2*x^(1/3)]*Cos[1/2*3^(1/2)* x^(1/3)]); CoefficientList[Series[f[x], {x, 0, max}], x]*(3 Range[0, max])! (* _Jean-François Alcover_, Sep 16 2013, after _Vladeta Jovovic_ *)
%Y Cf. A000364, A178963, A278073.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Feb 13 2005
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