A157805 Numerator of Euler(n,3).

1, 5, 6, 55, 30, 125, 126, 2015, 510, 2075, 2046, 15685, 8190, 38225, 32766, 118975, 131070, 3726575, 524286, -217736285, 2097150, 4730505125, 8388606, -968249463115, 33554430, 14717801331875, 134217726, -2093659805510855, 536870910, 86125674710684825

Offset: 0

Author

N. J. A. Sloane, Nov 10 2009

Keywords

• sign
• frac

Examples

			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
		

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.

Crossrefs

For denominators see A006519.

Cf. A002425.

Programs

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 *)

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 an 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)