A238800 Unreduced numerators in triangle that leads to the Euler numbers A198631(n)/A006519(n+1).

1, 1, 1, -2, 1, -3, 1, -4, 2, 1, -5, 5, 1, -6, 9, -10, 1, -7, 14, -35, 1, -8, 20, -80, 26, 1, -9, 27, -150, 117, 1, -10, 35, -250, 325, -454, 1, -11, 44, -385, 715, -2497, 1, -12, 54, -560, 1365, -8172, 5914, 1, -13

Offset: 0

Author

Paul Curtz, Mar 05 2014

Keywords

• sign
• tabf

Comments

We use the array ASPEC mentioned in A191302:

2, 1, 1, 1, 1, 1, 1, 1,...

2, 3, 4, 5, 6, 7, 8, 9,...

2, 5, 9, 14, 20, 27, 35, 44,...

2, 7, 16, 30, 50, 77, 112, 156,...

with the first upper diagonal of the difference table of the autosequence A198631(n)/A006519(n+1), i.e., 1/2, -1/4, 1/4, -5/8, 13/4, -227/8, 2957/8,...

written by columns:

1/2

1/2,

1/2, -1/4,

1/2, -1/4,

1/2, -1/4, 1/4,

1/2, -1/4, 1/4,

1/2, -1/4, 1/4, -5/8,

1/2, -1/4, 1/4, -5/8,

etc.

Hence, by multiplication of this double triangle by ASPEC, the beginning of the double triangle ESPEC is obtained:

E(0) = 1 = 1

E(1) = 1/2 = 1/2

E(2) = 0 = 1/2 -2/4

E(3) = -1/4 = 1/2 -3/4

E(4) = 0 = 1/2 -4/4 +2/4

E(5) = 1/2 = 1/2 -5/4 +5/4

E(6) = 0 = 1/2 -6/4 +9/4 -10/8

E(7) = -17/8 = 1/2 -7/4 +14/4 -35/8

E(8) = 0 = 1/2 -8/4 +20/4 -80/8 +26/4.

The terms of the sequence are the reduced numerators. Like A192456(n) for Bernoulli numbers A164555(n)/A027642(n).

Examples

			a(n) by triangle
1,
1,
1, -2,
1, -3,
1, -4,  2,
1, -5,  5,
1, -6,  9, -10,
1, -7, 14, -35,
1, -8, 20, -80, 26,
etc.