A217714 Modified Euler numbers.

1, 0, -2, -3, 4, 15, -62, -273, 1384, 7935, -50522, -353793, 2702764, 22368255, -199360982, -1903757313, 19391512144, 209865342975, -2404879675442, -29088885112833, 370371188237524, 4951498053124095, -69348874393137902, -1015423886506852353, 15514534163557086904, 246921480190207983615, -4087072509293123892362

Offset: 0

Author

Paul Curtz, Mar 21 2013

Keywords

• sign

Comments

a(n) and differences are:

1, 0, -2, -3, 4, 15, -62;

-1, -2, -1, 7, 11, -77;

-1, 1, 8, 4, -88;

2, 7, -4, -92;

5, -11, -88;

-16, -77;

-61;

(See the array in A163982(n) and the comments/examples in A090158 and A090145.)

The absolute values of the first column are A000111(n).

The first column can be found via the Akiyama-Tanigawa algorithm. See the chapter on the Seidel triangle in Wikipedia's Bernoulli Number.

Examples

			a(0) =   1;
a(1) =   1 -   1 = 0;
a(2) =  -1 -   2 +   1 = -2;
a(3) =   2 -   3 -   3 +   1 = -3;
a(4) =   5 +   8 -   6 -   4 +  1 = 4;
a(5) = -16 +  25 +  20 -  10 -  5 +  1 = 15;
a(6) = -61 -  96 +  75 +  40 - 15 -  6 + 1 = -62;
a(7) = 272 - 427 - 336 + 175 + 70 - 21 - 7 + 1 = -273; - _Philippe Deléham_, Oct 27 2013
G.f. = 1 - 2*x^2 - 3*x^3 + 4*x^4 + 15*x^5 - 62*x^6 - 273*x^7 + ...
		

Links

• Wikipedia, Bernoulli Number, Seidel triangle

Crossrefs

Cf. A000111, A163982.

Programs

Mathematica

a[n_] := 2^n* EulerE[n, 1] + EulerE[n] - 1; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Mar 21 2013 *)

Formula

a(n) = -A163982(n) - 1.

a(n) = Sum_{k=0..n} A109449(n,k)*floor((n-k+1)/2). - Philippe Deléham, Oct 27 2013

E.g.f.: 1/cosh(x) + tanh(x) + 1 - exp(x). - Sergei N. Gladkovskii, Nov 10 2014

Extensions

More terms from Jean-François Alcover, Mar 21 2013