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A240559 a(n) = -2^n*(E(n, 1/2) + E(n, 1) + (n mod 2)*2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

%I A240559 #54 Feb 22 2021 02:52:55
%S A240559 0,0,1,-3,-5,45,61,-1113,-1385,42585,50521,-2348973,-2702765,
%T A240559 176992725,199360981,-17487754833,-19391512145,2195014332465,
%U A240559 2404879675441,-341282303124693,-370371188237525,64397376340013805,69348874393137901,-14499110277050234553
%N A240559 a(n) = -2^n*(E(n, 1/2) + E(n, 1) + (n mod 2)*2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.
%H A240559 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.
%F A240559 a(2*n) = (-1)^(n+1)*A147315(2*n,1) = (-1)^(n+1)*A186370(2*n,2*n) =(-1)^(n+1)*A000364(n) for n>0.
%F A240559 a(2*n+1) = (-1)^n*A147315(2*n+1,2) = (-1)^n*A186370(2*n,2*n-1) = A241242(n).
%F A240559 a(n) = Sum_{k=0..n} (-1)^(n-k)*2^k*binomial(n,k)*(E(k,1/2) + 2*E(k+1,0)) where E(n,x) are the Euler polynomials.
%F A240559 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*(skp(k,0) + skp(k+1,-1)), where skp(n, x) are the Swiss-Knife polynomials A153641.
%F A240559 a(n) = A239322(n) + A239005(n+1) - A239005(n). - _Paul Curtz_, Apr 18 2014
%F A240559 E.g.f.: 1 - sech(x) - tanh(x) + sinh(x)*sech(x)^2 = ((exp(-x)-1)*sech(x))^2 / 2. - _Sergei N. Gladkovskii_, Nov 20 2014
%F A240559 E.g.f.: (1 - sech(x)) * (1 - tanh(x)). - _Michael Somos_, Nov 22 2014
%e A240559 G.f. = x^2 - 3*x^3 - 5*x^4 + 45*x^5 + 61*x^6 - 1113*x^7 - 1385*x^8 + ...
%p A240559 A240559 := proc(n) euler(n,1/2) + euler(n,1); if n mod 2 = 1 then % + 2*(euler(n+1,1/2)+euler(n+1,1)) fi; -2^n*% end: seq(A240559(n),n=0..19);
%t A240559 skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; skp[n_, x0_?NumericQ] := skp[n, x] /. x -> x0; a[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*(skp[k, 0] + skp[k+1, -1]), {k, 0, n}]; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Dec 09 2014, after _Peter Luschny_ *)
%o A240559 (Sage)
%o A240559 # Efficient computation with L. Seidel's boustrophedon transformation.
%o A240559 def A240559_list(n) :
%o A240559     A = [0]*(n+1); A[0] = 1; R = [0]
%o A240559     k = 0; e = 1; x = -1; s = -1
%o A240559     for i in (0..n):
%o A240559         Am = 0; A[k + e] = 0; e = -e;
%o A240559         for j in (0..i): Am += A[k]; A[k] = Am; k += e
%o A240559         if e == 1: x += 1; s = -s
%o A240559         v = -A[-x] if e == 1 else A[-x] - A[x]
%o A240559         if i > 1: R.append(s*v)
%o A240559     return R
%o A240559 A240559_list(24)
%Y A240559 Cf. A000364, A147315, A186370, A241242.
%Y A240559 Cf. A239005, A239322.
%K A240559 sign
%O A240559 0,4
%A A240559 _Peter Luschny_, Apr 17 2014