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A240561 The main diagonal in the difference table of A240559.

%I A240561 #14 Apr 06 2015 21:31:08
%S A240561 0,1,-10,178,-5296,238816,-15214480,1301989648,-144118832896,
%T A240561 20040052293376,-3419989086092800,702831038438522368,
%U A240561 -171209091176316215296,48783404012394865985536,-16074763418934659189278720,6065554251200571899397081088,-2598468976240882751482797162496
%N A240561 The main diagonal in the difference table of A240559.
%F A240561 a(n) = -Sum_{k=0..n}(C(n,k)*Euler(n+k+1)). - _Vladimir Kruchinin_, Apr 06 2015
%F A240561 a(n) ~ (-1)^(n+1) * 2^(4*n+9/2) * n^(2*n+3/2) / (exp(2*n) * Pi^(2*n+3/2)). - _Vaclav Kotesovec_, Apr 06 2015
%e A240561 a(n) is the main diagonal in this difference table D(n, k):
%e A240561 [    0,     0,     1,    -3,    -5,    45,    61, -1113, -1385]
%e A240561 [    0,     1,    -2,    -8,    40,   106, -1052, -2498]
%e A240561 [    1,    -1,   -10,    32,   146,  -946, -3550]
%e A240561 [    0,   -11,    22,   178,  -800, -4496]
%e A240561 [  -11,    11,   200,  -622, -5296]
%e A240561 [    0,   211,  -422, -5918]
%e A240561 [  211,  -211, -6340]
%e A240561 [    0, -6551]
%e A240561 [-6551]
%e A240561 D(n, 0) = A240560(n).
%e A240561 D(0, n) = A240559(n).
%e A240561 D(2*n, 0) = (-1)^(n+1)*A147315(2*n, 2).
%p A240561 A240561_list := proc(len) local A, m, n, k;
%p A240561 n := 2*len-1; A := array(0..n, 0..n);
%p A240561 for m from 0 to n do
%p A240561    A[m, 0] := euler(m) + 2^(m+1)*euler(m+1,0);
%p A240561    for k from m-1 by -1 to 0 do
%p A240561       A[k, m-k] := A[k+1, m-k-1] - A[k, m-k-1]
%p A240561 od od; [seq(A[k, k], k=0..len-1)] end:
%p A240561 A240561_list(17);
%t A240561 Table[-Sum[Binomial[n, k]*EulerE[n+k+1], {k, 0, n}],{n,0,20}] (* _Vaclav Kotesovec_, Apr 06 2015 *)
%o A240561 (Maxima)
%o A240561 a(n):=-sum(binomial(n,k)*euler(n+k+1),k,0,n); /* _Vladimir Kruchinin_, Apr 06 2015 */
%Y A240561 Cf. A147315, A240559, A240560.
%K A240561 sign
%O A240561 0,3
%A A240561 _Peter Luschny_, Apr 17 2014