A166919 - OEIS

%I #16 Jun 09 2022 14:41:51

%S 1,-1,2,1,-1,-6,-5,2,3,1,-1,24,26,-3,-14,-13,-2,3,3,1,-1,-120,-154,

%T -11,73,79,47,13,-21,-22,-9,-1,3,3,1,-1,720,1044,220,-427,-547,-361,

%U -245,-41,142,149,94,30,-8,-30,-17,-8,-1,3,3,1,-1

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

%H G. C. Greubel, <a href="/A166919/b166919.txt">Rows n = 0..30 of the irregular triangle, flattened</a>

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

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

%F T(n, binomial(n+1,2) - 1) = -1. - _G. C. Greubel_, Mar 27 2022

%e Irregular triangle begins as:

%e      1;

%e     -1;

%e      2,    1,  -1;

%e     -6,   -5,   2,   3,   1, -1;

%e     24,   26,  -3, -14, -13, -2,  3,   3,   1, -1;

%e   -120, -154, -11,  73,  79, 47, 13, -21, -22, -9, -1, 3, 3, 1, -1;

%t (* First program *)

%t p[n_, x_]:= p[n, x]= Product[-k-x +x^k, {k, n}];

%t Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten

%t (* Second program *)

%t m:=11;

%t T[n_, k_]:= T[n, k]= Coefficient[Series[Product[-j-x +x^j, {j, n}], {x, 0, Binomial[m+1,2]}], x, k];

%t Join[{1}, Table[T[n, k], {n,m}, {k,0,Binomial[n+1,2] -1}]//Flatten] (* _G. C. Greubel_, Mar 27 2022 *)

%o (Sage)

%o def p(n,x): return product(x^j -x-j for j in (1..n))

%o def A166919(n,k): return ( p(n,x) ).series(x, binomial(n+1,2)).list()[k]

%o [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

%K sign,tabf

%O 0,3

%A _Roger L. Bagula_, Oct 23 2009

%E Edited by _G. C. Greubel_, Mar 27 2022