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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.
1
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
0,1
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, ... .
FORMULA
a(n) = A007318(n) - A197870(n+1).
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 *)
KEYWORD
sign,tabl
AUTHOR
Paul Curtz, Oct 23 2016
STATUS
approved