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 A002115 Generalized Euler numbers. (Formerly M5082 N2199) 14
 1, 1, 19, 1513, 315523, 136085041, 105261234643, 132705221399353, 254604707462013571, 705927677520644167681, 2716778010767155313771539, 14050650308943101316593590153, 95096065132610734223282520762883, 823813936407337360148622860507620561 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES 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). LINKS 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. FORMULA 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 MAPLE 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 MATHEMATICA 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 *) CROSSREFS Cf. A000364, A178963, A278073. Sequence in context: A177611 A051847 A217830 * A223498 A054949 A242564 Adjacent sequences:  A002112 A002113 A002114 * A002116 A002117 A002118 KEYWORD nonn AUTHOR EXTENSIONS More terms from Vladeta Jovovic, Feb 13 2005 STATUS approved

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Last modified April 18 01:48 EDT 2021. Contains 343072 sequences. (Running on oeis4.)