OFFSET
0,2
LINKS
Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.
Vladimir Shevelev, A formula for numerator of Euler(n,k), Wed Sep 06 2017.
FORMULA
From Vladimir Shevelev, Sep 04 2017: (Start)
For even n, a(n) = 2^(n+1) - 2 + delta(n,0), where delta is the Kronecker symbol; for n == 1 (mod 4), a(n) = (2^(n+1)-2)*A006519(n+1) + A002425((n+1)/2); for n == 3 (mod 4), a(n) = (2^(n+1)-2)*A006519(n+1) - A002425((n+1)/2).
A generalization: Let N(n,k) denote numerator of Euler(n,k), k >= 1 is integer. Set u(n,k) = 2*Sum_{1 <= i <= k-1}(-1)^(i-1)*(k-i)^n. Then, for even n, N(n,k) = u(n,k) + (-1)^(k-1)^delta(n,0); for n == 1 (mod 4), N(n,k) = u(n,k)*A006519(n+1) + (-1)^(k-1)*A002425((n+1)/2); for n == 3 (mod 4), N(n,k) = u(n,k)* A006519 (n+1) - (-1)^(k-1)*A002425((n+1)/2). (End)
EXAMPLE
By the formula, we have a(1) = 2*2 + 1 = 5, a(3) = 14*4 - 1 = 55, a(5) = 62*2 + 1 = 125, a(7) = 254*8 - 17 = 2015, a(9) = 1022*2 + 31 = 2075, etc. - Vladimir Shevelev, Sep 04 2017
MATHEMATICA
a2425[n_] := (-1)^n/n*(1 - 4^n)*BernoulliB[2*n]*2^IntegerExponent[2*n, 2];
a6519[n_] := 2^IntegerExponent[n, 2];
a[n_] := Switch[Mod[n, 4], 0 | 2, 2^(n + 1) - 2 + KroneckerDelta[n, 0], 1, (2^(n + 1) - 2)*a6519[n + 1] + a2425[(n + 1)/2], 3, (2^(n + 1) - 2)*a6519[n + 1] - a2425[(n + 1)/2]];
Table[a[n], {n, 0, 30}]
(* or *)
Table[EulerE[n, 3] // Numerator, {n, 0, 30}] (* Jean-François Alcover, Jul 14 2018 *)
CROSSREFS
KEYWORD
sign,frac
AUTHOR
N. J. A. Sloane, Nov 10 2009
STATUS
approved