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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-Fran├žois 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

N. J. A. Sloane

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.)