A156578 Triangle of coefficients of 1 - (n+1)*x^n + n*x^(n+1), read by rows.

0, 1, -2, 1, 1, 0, -3, 2, 1, 0, 0, -4, 3, 1, 0, 0, 0, -5, 4, 1, 0, 0, 0, 0, -6, 5, 1, 0, 0, 0, 0, 0, -7, 6, 1, 0, 0, 0, 0, 0, 0, -8, 7, 1, 0, 0, 0, 0, 0, 0, 0, -9, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, -10, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 10

Offset

0,3

Links

G. C. Greubel, Rows n = 0..50 if the irregular triangle, flattened

Formula

T(n, k) = [x^k]( (1-x)^2 * Sum_{j=0..n-1} (j+1)*x^j ).

T(n, k) = [k=0] - (n+1)*[k=n] + n*[k=n+1] for n > 0, with T(0, 0) = 0. - G. C. Greubel, Jan 03 2022

Example

Irregular triangle begins as:
  0;
  1, -2,  1;
  1,  0, -3,  2;
  1,  0,  0, -4,  3;
  1,  0,  0,  0, -5,  4;
  1,  0,  0,  0,  0, -6,  5;
  1,  0,  0,  0,  0,  0, -7,  6;
  1,  0,  0,  0,  0,  0,  0, -8,  7;
  1,  0,  0,  0,  0,  0,  0,  0, -9,   8;
  1,  0,  0,  0,  0,  0,  0,  0,  0, -10,   9;
  1,  0,  0,  0,  0,  0,  0,  0,  0,   0, -11, 10;
  ...

Mathematica

T[n_]:= If[n==0, 0, CoefficientList[1 -(n+1)*x^n +n*x^(n+1), x]];
Table[T[n], {n, 0, 15}]//Flatten (* modified by G. C. Greubel, Jan 03 2022 *)

Prog

(Sage) [0]+flatten([[( 1 -(n+1)*x^n +n*x^(n+1) ).series(x, n+2).list()[k] for k in (0..n+1)] for n in (1..12)]) # G. C. Greubel, Jan 03 2022

Crossrefs

Sequence in context: A340011 A055168 A085144 * A171846 A097230 A144789

Adjacent sequences: A156575 A156576 A156577 * A156579 A156580 A156581

Keyword

sign,tabf,less

Author

Roger L. Bagula, Feb 10 2009

Extensions

Edited by G. C. Greubel, Jan 03 2022

Status

approved