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

%I A099028 #17 Mar 07 2020 08:50:43
%S A099028 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 A099028 -96,-96,-427,-427,-402,-352,-280,-192,-96,0,0,427,854,1256,1608,1888,
%U A099028 2080,2176,2176,12465,12465,12038,11184,9928,8320,6432,4352,2176,0
%N A099028 Euler-Seidel matrix T(k,n) with start sequence e.g.f. 2x/(1+e^(2x)), read by antidiagonals.
%C A099028 In an Euler-Seidel matrix, the rows are consecutive pairwise sums and the columns consecutive differences.
%H A099028 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 A099028 Recurrence: T(k, n) = T(k-1, n) + T(k-1, n+1).
%e A099028 Seidel matrix:
%e A099028 [    0     1    -2     0     8     0   -96     0  2176     0]
%e A099028 [    1    -1    -2     8     8   -96   -96  2176  2176     .]
%e A099028 [    0    -3     6    16   -88  -192  2080  4352     .     .]
%e A099028 [   -3     3    22   -72  -280  1888  6432     .     .     .]
%e A099028 [    0    25   -50  -352  1608  8320     .     .     .     .]
%e A099028 [   25   -25  -402  1256  9928     .     .     .     .     .]
%e A099028 [    0  -427   854 11184     .     .     .     .     .     .]
%e A099028 [ -427   427 12038     .     .     .     .     .     .     .]
%e A099028 [    0 12465     .     .     .     .     .     .     .     .]
%e A099028 [12465     .     .     .     .     .     .     .     .     .]
%t A099028 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 A099028 Table[T[k-n, n], {k, 0, 9}, {n, 0, k}] (* _Jean-François Alcover_, Jun 11 2019 *)
%o A099028 (Sage)
%o A099028 def SeidelMatrixA099028(dim):
%o A099028     E = matrix(ZZ, dim)
%o A099028     t = taylor(2*x/(1+exp(2*x)), x, 0, dim + 1)
%o A099028     for k in (0..dim-1):
%o A099028         E[0, k] = factorial(k) * t.coefficient(x, k)
%o A099028     R = [0]
%o A099028     for n in (1..dim-1):
%o A099028         for k in (0..dim-n-1):
%o A099028             E[n, k] = E[n-1, k] + E[n-1, k+1]
%o A099028         R.extend([E[n-k,k] for k in (0..n)])
%o A099028     return R
%o A099028 print(SeidelMatrixA099028(10)) # _Peter Luschny_, Jul 02 2016
%Y A099028 First column (odd part) is A009843, main diagonal is in A099029. Antidiagonal sums are in A065619. Cf. A009752.
%K A099028 sign,tabl
%O A099028 0,6
%A A099028 _Ralf Stephan_, Sep 27 2004