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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 (list; table; graph; refs; listen; history; text; internal format)
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, ... .

LINKS

Table of n, a(n) for n=0..65.

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-Fran├žois Alcover, Nov 16 2016 *)

CROSSREFS

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

Sequence in context: A090723 A027357 A256479 * A283715 A078808 A030363

Adjacent sequences:  A277075 A277076 A277077 * A277079 A277080 A277081

KEYWORD

sign,tabl

AUTHOR

Paul Curtz, Oct 23 2016

STATUS

approved

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Last modified September 27 19:34 EDT 2021. Contains 347694 sequences. (Running on oeis4.)