A277078 - OEIS

%I #57 Nov 30 2016 06:15:40

%S 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 2,1,-7,21,-35,35,-21,7,0,1,-8,28,-56,70,-56,28,-8,2,1,-9,36,-84,126,

%U -126,84,-36,9,0,1,-10,45,-120,210,-252,210,-120,45,-10,2

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

%C a(n)=

%C 2,

%C 1,  0,

%C 1, -2, 2,

%C 1, -3, 3,  0,

%C 1, -4, 6, -4, 2,

%C etc.

%C transforms every sequence s(n) in an autosequence of the second kind via the multiplication by the triangle

%C s0,           T2

%C s0, s1,

%C s0, s1, s2,

%C s0, s1, s2, s3,

%C etc.

%C which is the reluctant form of s(n).

%C Example.

%C s(n) = A131577(n) = 0, 1, 2, 4, ... .

%C The multiplication gives 0, 0, 2, 3, 8, 15, 32, 63, ... = 0 followed by A166920.

%C 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 a(n) = A007318(n) - A197870(n+1).

%t 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 Cf. A000035, A007318, A054977, A131577, A166920, A197870, A255935.

%K sign,tabl

%O 0,1

%A _Paul Curtz_, Oct 23 2016