A099028 Euler-Seidel matrix T(k,n) with start sequence e.g.f. 2x/(1+e^(2x)), read by antidiagonals.

0, 1, 1, 0, -1, -2, -3, -3, -2, 0, 0, 3, 6, 8, 8, 25, 25, 22, 16, 8, 0, 0, -25, -50, -72, -88, -96, -96, -427, -427, -402, -352, -280, -192, -96, 0, 0, 427, 854, 1256, 1608, 1888, 2080, 2176, 2176, 12465, 12465, 12038, 11184, 9928, 8320, 6432, 4352, 2176, 0

Offset: 0

Author

Ralf Stephan, Sep 27 2004

Keywords

• sign
• tabl

Comments

In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences.

Examples

			Seidel matrix:
[    0     1    -2     0     8     0   -96     0  2176     0]
[    1    -1    -2     8     8   -96   -96  2176  2176     .]
[    0    -3     6    16   -88  -192  2080  4352     .     .]
[   -3     3    22   -72  -280  1888  6432     .     .     .]
[    0    25   -50  -352  1608  8320     .     .     .     .]
[   25   -25  -402  1256  9928     .     .     .     .     .]
[    0  -427   854 11184     .     .     .     .     .     .]
[ -427   427 12038     .     .     .     .     .     .     .]
[    0 12465     .     .     .     .     .     .     .     .]
[12465     .     .     .     .     .     .     .     .     .]
		

Links

• D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Crossrefs

First column (odd part) is A009843, main diagonal is in A099029. Antidiagonal sums are in A065619. Cf. A009752.

Programs

Mathematica

T[k_, n_] := T[k, n] = If[k == 0, SeriesCoefficient[2x/(1 + E^(2x)), {x, 0, n}] n!, T[k-1, n] + T[k-1, n+1]];
Table[T[k-n, n], {k, 0, 9}, {n, 0, k}] (* Jean-François Alcover, Jun 11 2019 *)

Sage

def SeidelMatrixA099028(dim):
    E = matrix(ZZ, dim)
    t = taylor(2*x/(1+exp(2*x)), x, 0, dim + 1)
    for k in (0..dim-1):
        E[0, k] = factorial(k) * t.coefficient(x, k)
    R = [0]
    for n in (1..dim-1):
        for k in (0..dim-n-1):
            E[n, k] = E[n-1, k] + E[n-1, k+1]
        R.extend([E[n-k,k] for k in (0..n)])
    return R
print(SeidelMatrixA099028(10)) # Peter Luschny, Jul 02 2016

Formula

Recurrence: T(k, n) = T(k-1, n) + T(k-1, n+1).