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A281588 E.g.f. z*(2*(exp(z) + 1)/(cosh(z) + cos(z)) - 1).

%I A281588 #21 Feb 22 2021 02:53:16
%S A281588 0,1,2,3,4,-5,-24,-98,-272,621,4960,31856,132672,-437593,-4893056,
%T A281588 -42854160,-237969664,1026405753,14756156928,163699919104,
%U A281588 1136284574720,-6054175060941,-106379840985088,-1428593836836352,-11899498670002176,75477454065058725
%N A281588 E.g.f. z*(2*(exp(z) + 1)/(cosh(z) + cos(z)) - 1).
%H A281588 L. Seidel, <a href="http://publikationen.badw.de/de/003384831/pdf/CC%20BY">Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen</a>, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), 157-187.
%p A281588 A281588_list := proc(n) z*(2*(exp(z)+1)/(cosh(z)+cos(z))-1);
%p A281588 series(%,z,n+1); seq(k!*coeff(%,z,k),k=0..n) end: A281588_list(25);
%o A281588 (Sage)
%o A281588 def SIB(dim, m): # Algorithm of L. Seidel (1877).
%o A281588     E = matrix(ZZ, dim)
%o A281588     def plow(n, dir):
%o A281588         if dir : # right to left backward
%o A281588             E[n, 0] = int(n == 1)
%o A281588             for k in range(n-1, -1, -1) :
%o A281588                 E[k, n-k] = E[k+1, n-k-1] - E[k, n-k-1]
%o A281588         else:  # left to right forward
%o A281588             E[0, n] = 0
%o A281588             for k in range(1, n+1, 1) :
%o A281588                 E[k, n-k] = E[k-1, n-k+1] + E[k-1, n-k]
%o A281588     dir = [mod(n, m) == 1 for n in (0..dim-1)]
%o A281588     for n in (0..dim-1): plow(n, dir[n])
%o A281588     return [E[0,k] if dir[k] else E[k,0] for k in range(dim)]
%o A281588 A281588_list = lambda len: SIB(len, 4)
%o A281588 A281588_list(26)
%Y A281588 Cf. A281587.
%K A281588 sign
%O A281588 0,3
%A A281588 _Peter Luschny_, Feb 01 2017