A002115 Generalized Euler numbers.

1, 1, 19, 1513, 315523, 136085041, 105261234643, 132705221399353, 254604707462013571, 705927677520644167681, 2716778010767155313771539, 14050650308943101316593590153, 95096065132610734223282520762883, 823813936407337360148622860507620561

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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
• Takao Komatsu and Ram Krishna Pandey, On hypergeometric Cauchy numbers of higher grade, AIMS Mathematics (2021) Vol. 6, Issue 7, 6630-6646.
• Takao Komatsu and Guo-Dong Liu, Congruence properties of Lehmer-Euler numbers, arXiv:2501.01178 [math.NT], 2025.
• D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
• Bruce E. Sagan, Generalized Euler numbers and ordered set partitions, arXiv:2501.07692 [math.NT], 2025. See p. 3.

Crossrefs

Cf. A000364, A178963, A278073.

Programs

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
# Alternative:
h := 1 / hypergeom([], [1/3, 2/3], (-x/3)^3): ser := series(h, x, 40):
seq((3*n)! * coeff(ser, x, 3*n), n = 0..13); # Peter Luschny, Mar 13 2023

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 *)

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). - 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

a(n) = (3*n)! * [x^(3*n)] hypergeom([], [1/3, 2/3], (-x/3)^3)^(-1). - Peter Luschny, Mar 13 2023

Extensions

More terms from Vladeta Jovovic, Feb 13 2005