A240561 The main diagonal in the difference table of A240559.

0, 1, -10, 178, -5296, 238816, -15214480, 1301989648, -144118832896, 20040052293376, -3419989086092800, 702831038438522368, -171209091176316215296, 48783404012394865985536, -16074763418934659189278720, 6065554251200571899397081088, -2598468976240882751482797162496

Offset: 0

Author

Peter Luschny, Apr 17 2014

Keywords

• sign

Examples

			a(n) is the main diagonal in this difference table D(n, k):
[    0,     0,     1,    -3,    -5,    45,    61, -1113, -1385]
[    0,     1,    -2,    -8,    40,   106, -1052, -2498]
[    1,    -1,   -10,    32,   146,  -946, -3550]
[    0,   -11,    22,   178,  -800, -4496]
[  -11,    11,   200,  -622, -5296]
[    0,   211,  -422, -5918]
[  211,  -211, -6340]
[    0, -6551]
[-6551]
D(n, 0) = A240560(n).
D(0, n) = A240559(n).
D(2*n, 0) = (-1)^(n+1)*A147315(2*n, 2).
		

Crossrefs

Cf. A147315, A240559, A240560.

Programs

Maple

A240561_list := proc(len) local A, m, n, k;
n := 2*len-1; A := array(0..n, 0..n);
for m from 0 to n do
   A[m, 0] := euler(m) + 2^(m+1)*euler(m+1,0);
   for k from m-1 by -1 to 0 do
      A[k, m-k] := A[k+1, m-k-1] - A[k, m-k-1]
od od; [seq(A[k, k], k=0..len-1)] end:
A240561_list(17);

Mathematica

Table[-Sum[Binomial[n, k]*EulerE[n+k+1], {k, 0, n}],{n,0,20}] (* Vaclav Kotesovec, Apr 06 2015 *)

Maxima

a(n):=-sum(binomial(n,k)*euler(n+k+1),k,0,n); /* Vladimir Kruchinin, Apr 06 2015 */

Formula

a(n) = -Sum_{k=0..n}(C(n,k)*Euler(n+k+1)). - Vladimir Kruchinin, Apr 06 2015

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