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

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, 2, 1, -7, 21, -35, 35, -21, 7, 0, 1, -8, 28, -56, 70, -56, 28, -8, 2, 1, -9, 36, -84, 126, -126, 84, -36, 9, 0, 1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 2

Offset: 0

Author

Paul Curtz, Oct 23 2016

Keywords

• sign
• tabl

Comments

a(n)=

2,

1, 0,

1, -2, 2,

1, -3, 3, 0,

1, -4, 6, -4, 2,

etc.

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

s0, T2

s0, s1,

s0, s1, s2,

s0, s1, s2, s3,

etc.

which is the reluctant form of s(n).

Example.

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

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

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, ... .

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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