A166919 Irregular triangle of coefficients of Product_{j=1..n} (x^j - x - 1), read by rows.

1, -1, 2, 1, -1, -6, -5, 2, 3, 1, -1, 24, 26, -3, -14, -13, -2, 3, 3, 1, -1, -120, -154, -11, 73, 79, 47, 13, -21, -22, -9, -1, 3, 3, 1, -1, 720, 1044, 220, -427, -547, -361, -245, -41, 142, 149, 94, 30, -8, -30, -17, -8, -1, 3, 3, 1, -1

Offset

0,3

Links

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

Formula

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = Product_{j=1..n} (-j - x + x^j).

T(n, 0) = (-1)^n * n!.

T(n, binomial(n+1,2) - 1) = -1. - G. C. Greubel, Mar 27 2022

Example

Irregular triangle begins as:
     1;
    -1;
     2,    1,  -1;
    -6,   -5,   2,   3,   1, -1;
    24,   26,  -3, -14, -13, -2,  3,   3,   1, -1;
  -120, -154, -11,  73,  79, 47, 13, -21, -22, -9, -1, 3, 3, 1, -1;

Mathematica

(* First program *)
p[n_, x_]:= p[n, x]= Product[-k-x +x^k, {k, n}];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten
(* Second program *)
m:=11;
T[n_, k_]:= T[n, k]= Coefficient[Series[Product[-j-x +x^j, {j, n}], {x, 0, Binomial[m+1, 2]}], x, k];
Join[{1}, Table[T[n, k], {n, m}, {k, 0, Binomial[n+1, 2] -1}]//Flatten] (* G. C. Greubel, Mar 27 2022 *)

Prog

(Sage)
def p(n, x): return product(x^j -x-j for j in (1..n))
def A166919(n, k): return ( p(n, x) ).series(x, binomial(n+1, 2)).list()[k]
[1]+flatten([[A166919(n, k) for k in (0..binomial(n+1, 2)-1)] for n in (1..10)]) # G. C. Greubel, Mar 27 2022

Crossrefs

Sequence in context: A330490 A199063 A140956 * A338874 A338876 A260238

Adjacent sequences: A166916 A166917 A166918 * A166920 A166921 A166922

Keyword

sign,tabf

Author

Roger L. Bagula, Oct 23 2009

Extensions

Edited by G. C. Greubel, Mar 27 2022

Status

approved