A123226 Irregular triangle formed by coefficients of the polynomials p(k,x) = (x^k - Sum_{j=0..k-1} x^j )*p(k - 1, x), with p(0,x)=1, p(1,x)=x+1, read by rows.

1, 1, 1, -1, -2, 0, 1, 1, 3, 3, 0, -3, -1, 1, -1, -4, -7, -7, -2, 4, 6, 3, -3, -2, 1, 1, 5, 12, 19, 21, 15, 2, -11, -15, -10, -1, 7, 7, -2, -3, 1, -1, -6, -18, -37, -58, -73, -73, -53, -19, 17, 41, 43, 25, 3, -13, -19, -11, 4, 11, 0, -4, 1

Offset

1,5

Links

G. C. Greubel, Rows n=0..30 of triangle, flattened

Formula

Equals coefficients of the polynomials p(k, x) = - ((1 - 2*x^k + x^(k+1))/(1-x))*p(k-1, x), with p(0, x) = 1, p(1, x) = x+1. - G. C. Greubel, Mar 23 2019

Example

Irregular triangle of coefficients begins as:
   1;
   1,  1;
  -1, -2,  0,  1;
   1,  3,  3,  0, -3, -1, 1;
  -1, -4, -7, -7, -2,  4, 6, 3, -3, -2, 1; ...

Mathematica

p[0, x] = 1; p[1, x] = x + 1;
p[k_, x_]:= p[k, x] = (x^k - Sum[x^m, {m, 0, k-1}])*p[k-1, x];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten

Crossrefs

Cf. A008302.

Sequence in context: A293108 A172237 A246181 * A347382 A299919 A238270

Adjacent sequences: A123223 A123224 A123225 * A123227 A123228 A123229

Keyword

tabf,sign,less

Author

Roger L. Bagula, Oct 05 2006

Status

approved