A379781 a(1)=1, a(2)=2; thereafter, a(n) is the final value at the bottom of the difference triangle of the sequence thus far.

1, 2, 1, -2, 0, 7, -14, -12, 155, -408, -364, 7693, -30940, 10712, 637701, -4224222, 9980180, 61922567, -810337234, 4100137008, -958593005, -174952472228, 1662063951016, -6944673371867, -22887336602200, 655644589917172, -5694691183524699, 19946666531550638, 176993602416669640

Offset: 1

Author

Neal Gersh Tolunsky, Jan 02 2025

Keywords

• sign

Comments

The difference triangle refers to the triangular array of iterated differences.

The first term in each row of the difference triangle is the inverse binomial transform of the sequence, so the definition means the inverse binomial transform deletes term a(2) = 2.

Examples

			To find a(6), we look at the first difference triangle of the first 5 terms:
  1,   2,   1,  -2,   0
  1,  -1,  -3,   2
 -2,  -2,   5
  0,   7
  7
7 is the final value, so a(6)=7.
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..595

Crossrefs

Cf. A014182, A010739.

Programs

Mathematica

a[1] = 1; a[2] = 2; a[n_] := a[n] = Sum[(-1)^(n-k+1) * Binomial[n-2, k-1] * a[k], {k, 1, n-1}]; Table[a[n], {n, 1, 30}] (* Amiram Eldar, Jan 04 2025 *)