OFFSET
0,6
COMMENTS
In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences.
LINKS
D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
FORMULA
Recurrence: T(k, n) = T(k-1, n) + T(k-1, n+1).
EXAMPLE
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 . . . . . . . . .]
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 *)
PROG
(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
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Ralf Stephan, Sep 27 2004
STATUS
approved