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A002115 Generalized Euler numbers.
(Formerly M5082 N2199)
14

%I M5082 N2199

%S 1,1,19,1513,315523,136085041,105261234643,132705221399353,

%T 254604707462013571,705927677520644167681,2716778010767155313771539,

%U 14050650308943101316593590153,95096065132610734223282520762883,823813936407337360148622860507620561

%N Generalized Euler numbers.

%D D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.

%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>

%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

%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

%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-Fran├žois 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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Last modified November 20 14:54 EST 2019. Contains 329337 sequences. (Running on oeis4.)