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A002115
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Generalized Euler numbers.
(Formerly M5082 N2199)
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14
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1, 1, 19, 1513, 315523, 136085041, 105261234643, 132705221399353, 254604707462013571, 705927677520644167681, 2716778010767155313771539, 14050650308943101316593590153, 95096065132610734223282520762883, 823813936407337360148622860507620561
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OFFSET
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0,3
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..166
D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
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FORMULA
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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
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
Alternating row sums of A278073. - Peter Luschny, Sep 07 2017
a(n) = A178963(3n). - Alois P. Heinz, Aug 12 2019
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
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MAPLE
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1, `if`(t=0,
add(b(u-j, o+j-1, irem(t+1, 3)), j=1..u),
add(b(u+j-1, o-j, irem(t+1, 3)), j=1..o)))
end:
a:= n-> b(3*n, 0$2):
seq(a(n), n=0..17); # Alois P. Heinz, Aug 12 2019
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A000364, A178963, A278073.
Sequence in context: A177611 A051847 A217830 * A223498 A054949 A242564
Adjacent sequences: A002112 A002113 A002114 * A002116 A002117 A002118
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Vladeta Jovovic, Feb 13 2005
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STATUS
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approved
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