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Euler-Seidel matrix T(k,n) with start sequence e.g.f. 2x/(1+e^(2x)), read by antidiagonals.
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%I #17 Mar 07 2020 08:50:43

%S 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,

%T -96,-96,-427,-427,-402,-352,-280,-192,-96,0,0,427,854,1256,1608,1888,

%U 2080,2176,2176,12465,12465,12038,11184,9928,8320,6432,4352,2176,0

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

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

%H D. Dumont, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05dumont.html">Matrices d'Euler-Seidel</a>, Sem. Loth. Comb. B05c (1981) 59-78.

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

%e Seidel matrix:

%e [ 0 1 -2 0 8 0 -96 0 2176 0]

%e [ 1 -1 -2 8 8 -96 -96 2176 2176 .]

%e [ 0 -3 6 16 -88 -192 2080 4352 . .]

%e [ -3 3 22 -72 -280 1888 6432 . . .]

%e [ 0 25 -50 -352 1608 8320 . . . .]

%e [ 25 -25 -402 1256 9928 . . . . .]

%e [ 0 -427 854 11184 . . . . . .]

%e [ -427 427 12038 . . . . . . .]

%e [ 0 12465 . . . . . . . .]

%e [12465 . . . . . . . . .]

%t 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]];

%t Table[T[k-n, n], {k, 0, 9}, {n, 0, k}] (* _Jean-François Alcover_, Jun 11 2019 *)

%o (Sage)

%o def SeidelMatrixA099028(dim):

%o E = matrix(ZZ, dim)

%o t = taylor(2*x/(1+exp(2*x)), x, 0, dim + 1)

%o for k in (0..dim-1):

%o E[0, k] = factorial(k) * t.coefficient(x, k)

%o R = [0]

%o for n in (1..dim-1):

%o for k in (0..dim-n-1):

%o E[n, k] = E[n-1, k] + E[n-1, k+1]

%o R.extend([E[n-k,k] for k in (0..n)])

%o return R

%o print(SeidelMatrixA099028(10)) # _Peter Luschny_, Jul 02 2016

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

%K sign,tabl

%O 0,6

%A _Ralf Stephan_, Sep 27 2004