This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A240485 #25 Apr 10 2014 05:57:40
%S A240485 1,3,2,-1,-2,3,6,-17,-34,155,310,-2073,-4146,38227,76454,-929569,
%T A240485 -1859138,28820619,57641238,-1109652905,-2219305810,51943281731,
%U A240485 103886563462,-2905151042481,-5810302084962,191329672483963,382659344967926,-14655626154768697
%N A240485 a(n) = -Zeta(1-n)*n*(2^(n+1) - 4) - Zeta(-n)*(n+1)*(2^(n+2) - 2), for n = 0 the limit is understood.
%C A240485 Let G(m, n) denote the difference table of a(n):
%C A240485 1, 3, 2, -1, -2, 3, 6, -17, -34,...
%C A240485 2, -1, -3, -1, 5, 3, -23, -17,...
%C A240485 -3, -2, 2, 6, -2, -26, 6,...
%C A240485 1, 4, 4, -8, -24, 32,...
%C A240485 3, 0, -12, -16, 56,...
%C A240485 -3, -12, -4, 72,...
%C A240485 -9, 8, 76,...
%C A240485 17, 68,...
%C A240485 51,...
%C A240485 a(n) = G(0, n).
%C A240485 The main diagonal G(n, n) = 1, -1, 2, -8, 56, -608,... is essentially a signed version of A005439.
%C A240485 The first upper diagonal is the main diagonal multiplied by 3. G(n, n+1) = 3*G(n, n).
%C A240485 G(m, n) = G(m, n-1) + G(m+1,n-1).
%C A240485 Inverse binomial transform: after 1, 2, -3, A110501(n+1) is interleaved with 3*A110501(n+1), signed two by two. I. e. b(n) = 1, 2, -3, 1, 3, -3, -9, 17, 51,... . a(n+2) + b(n+2) = -1, 0, 1, 0, -3, 0, 17,... = A226158(n+2).
%C A240485 This is particular to the Genocchi numbers. If the first upper diagonal is proportional to the main diagonal (1, -1, 2, -8,...), the sequence and the inverse binomial transform are simply connected to the Genocchi numbers.
%F A240485 a(2*n+1) = a(2*n+2)/2 for n > 0.
%F A240485 -a(2*n+2)/2 = A226158(2*n+2) = A001469(n+1) = (2*n+2)*E(2*n+1, 0) where E(n, x) are the Euler polynomials.
%F A240485 a(n) = -2*A226158(n) - A226158(n+1).
%F A240485 E.g.f.: (2*exp(x)*(3*x+exp(x)*(2*x+1)+1))/(exp(x)+1)^2. - _Peter Luschny_, Apr 10 2014
%p A240485 A240485 := proc(n) if n = 0 then 1 elif n = 1 then 3 else
%p A240485 m := 2*iquo(n-1, 2) + 2; -2^irem(n-1, 2)*m*euler(m-1, 0) fi end:
%p A240485 seq(A240485(n), n=0..27); # _Peter Luschny_, Apr 09 2014
%t A240485 a[n_] := Which[n == 0, 1, n == 1, 3, True, m = 2*Quotient[n-1, 2]+2; -2^Mod[n-1, 2]*m*EulerE[m-1, 0]]; Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Apr 09 2014, after _Peter Luschny_ *)
%o A240485 (Sage)
%o A240485 def A240485(n):
%o A240485 if n < 3: return [1,3,2][n]
%o A240485 m = 2*((n+1)//2)
%o A240485 b = 2*(1-2^m)*bernoulli(m)
%o A240485 if is_even(n): b = 2*b
%o A240485 return (-1)^ceil((n^2+1)/2)*b
%o A240485 [A240485(n) for n in (0..24)] # _Peter Luschny_, Apr 08 2014
%Y A240485 Cf. A001469, A005439, A110501, A226158, A230324.
%K A240485 sign
%O A240485 0,2
%A A240485 _Paul Curtz_, Apr 06 2014