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
Keywords
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).
Programs
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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);
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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