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A277078 Triangular array similar to A255935 but with 0's and 2's swapped in the trailing diagonal. The columns alternate in signs.

%I A277078 #57 Nov 30 2016 06:15:40
%S A277078 2,1,0,1,-2,2,1,-3,3,0,1,-4,6,-4,2,1,-5,10,-10,5,0,1,-6,15,-20,15,-6,
%T A277078 2,1,-7,21,-35,35,-21,7,0,1,-8,28,-56,70,-56,28,-8,2,1,-9,36,-84,126,
%U A277078 -126,84,-36,9,0,1,-10,45,-120,210,-252,210,-120,45,-10,2
%N A277078 Triangular array similar to A255935 but with 0's and 2's swapped in the trailing diagonal. The columns alternate in signs.
%C A277078 a(n)=
%C A277078 2,
%C A277078 1,  0,
%C A277078 1, -2, 2,
%C A277078 1, -3, 3,  0,
%C A277078 1, -4, 6, -4, 2,
%C A277078 etc.
%C A277078 transforms every sequence s(n) in an autosequence of the second kind via the multiplication by the triangle
%C A277078 s0,           T2
%C A277078 s0, s1,
%C A277078 s0, s1, s2,
%C A277078 s0, s1, s2, s3,
%C A277078 etc.
%C A277078 which is the reluctant form of s(n).
%C A277078 Example.
%C A277078 s(n) = A131577(n) = 0, 1, 2, 4, ... .
%C A277078 The multiplication gives 0, 0, 2, 3, 8, 15, 32, 63, ... = 0 followed by A166920.
%C A277078 a(n) comes from alternate sum and difference of s(n) and t(n), its inverse binomial transform. In the example (t(n) = periodic 2: repeat 0, 1) the first terms are: 0+0, 1-1, 2+0, 4-1, 8+0, 16-1, 32+0, 64-1, ... .
%F A277078 a(n) = A007318(n) - A197870(n+1).
%t A277078 a[n_, k_] := If[k == n, 2 - 2*Mod[n, 2], (-1)^k*Binomial[n, k]]; Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 16 2016 *)
%Y A277078 Cf. A000035, A007318, A054977, A131577, A166920, A197870, A255935.
%K A277078 sign,tabl
%O A277078 0,1
%A A277078 _Paul Curtz_, Oct 23 2016