A239005 Signed version of the Seidel triangle for the Euler numbers, read by rows.

1, 0, 1, -1, -1, 0, 0, -1, -2, -2, 5, 5, 4, 2, 0, 0, 5, 10, 14, 16, 16, -61, -61, -56, -46, -32, -16, 0, 0, -61, -122, -178, -224, -256, -272, -272, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 0, 0, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936

Offset: 0

Author

Paul Curtz, Mar 08 2014

Keywords

• sign
• tabl

Examples

			The triangle T(n,k) begins:
                      1
                    0   1
                 -1  -1   0
                0  -1  -2  -2
              5   5   4   2   0
             ...
The array read as a table, A(n,k) = T(n+k, k), starts:
     1,    1,    0,   -2,    0,   16,    0, -272,    0, ...
     0,   -1,   -2,    2,   16,  -16, -272,  272, ...
    -1,   -1,    4,   14,  -32, -256,  544, ...
     0,    5,   10,  -46, -224,  800, ...
     5,    5,  -56, -178, 1024, ...
     0,  -61, -122, 1202, ...
   -61,  -61, 1324, ...
     0, 1385, ...
  1385, ...
  ...
For the above table, we have A(n,k) = (-1)^(n+k)*A236935(n,k) for n, k >= 0. It has joint e.g.f. 2*exp(-x)/(1 + exp(-2*(x+y))). - _Petros Hadjicostas_, Feb 21 2021
		

Links

• L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), pp. 157-187; see Beilage 4 (p. 187).

Crossrefs

Unsigned version is A008280.

Cf. A008281, A099023, A108040, A122045, A155585, A236935.

Programs

Mathematica

t[0, 0] = 1; t[n_, m_] /; nJean-François Alcover, Dec 30 2014 *)

Maxima

T(n,m):=sum(binomial(m,k)*euler(n-m+k),k,0,m); /* Vladimir Kruchinin, Apr 06 2015 */

PARI

a(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1) /* A122045 */
T(n, k) = (-1)^n*sum(i=0, k, (-1)^i*binomial(k, i)*a(n-i)) /* Petros Hadjicostas, Feb 21 2021 */
/* Second PARI program (same a(n) for A122045 as above) */
T(n, k) = sum(i=0, k, binomial(k, i)*a(n-k+i)) /* Petros Hadjicostas, Feb 21 2021 */

Formula

a(n) = A057077(n)*A008280(n) by rows.

a(n) is the increasing antidiagonals of the difference table of A155585(n).

Central column of triangle: A099023(n).

Right main diagonal of triangle: A155585(n) (see A009006(n)).

Left main diagonal of triangle: A122045(n).

T(n,m) = Sum_{k=0..n} binomial(m,k)*Euler(n-m+k) for 0 <= m <= n. - Vladimir Kruchinin, Apr 06 2015 [The summation only needs to go from k=0 to k=m because of binomial(m,k).]

T(n,k) = (-1)^n*A236935(n-k,k) for 0 <= k <= n, where the latter is read as a square array. - Petros Hadjicostas, Feb 21 2021