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A155124 Triangle T(n, k) = 1-n if k=0 else 2, read by rows. 2

%I

%S 1,0,2,-1,2,2,-2,2,2,2,-3,2,2,2,2,-4,2,2,2,2,2,-5,2,2,2,2,2,2,-6,2,2,

%T 2,2,2,2,2,-7,2,2,2,2,2,2,2,2,-8,2,2,2,2,2,2,2,2,2,-9,2,2,2,2,2,2,2,2,

%U 2,2,-10,2,2,2,2,2,2,2,2,2,2,2,-11,2,2,2,2,2,2,2,2,2,2,2,2

%N Triangle T(n, k) = 1-n if k=0 else 2, read by rows.

%C These polynomials in n are column functions for general Pascal-Sierpinski triangles.

%H G. C. Greubel, <a href="/A155124/b155124.txt">Rows n = 0..50 of the triangle, flattened</a>

%F Let p(x,m) = (1-m) + 2*x*(1-x^m)/(1-x) then the triangle is given by T(n, k) = coefficients( p(x,n) ).

%F Coefficient triangle of polynomial: p(x,n)= 1-m + 2*Sum_{k=1..n} x^k.

%F From _G. C. Greubel_, Mar 25 2021: (Start)

%F T(n, k) = 1-n if k=0 else 2.

%F Sum_{k=0..n} T(n ,k) = n+1 = A000027(n+1). (End)

%e Triangle begins as:

%e 1;

%e 0, 2;

%e -1, 2, 2;

%e -2, 2, 2, 2;

%e -3, 2, 2, 2, 2;

%e -4, 2, 2, 2, 2, 2;

%e -5, 2, 2, 2, 2, 2, 2;

%e -6, 2, 2, 2, 2, 2, 2, 2;

%e -7, 2, 2, 2, 2, 2, 2, 2, 2;

%e -8, 2, 2, 2, 2, 2, 2, 2, 2, 2;

%e -9, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;

%t Table[CoefficientList[-(m-1) + 2*x*(1-x^m)/(1-x), x], {m,0,15}]//Flatten

%t Table[If[k==0, 1-n, 2], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 25 2021 *)

%o (Magma) [k eq 0 select 1-n else 2: k in [0..n], n in [0..15]]; // _G. C. Greubel_, Mar 25 2021

%o (Sage) flatten([[1-n if k==0 else 2 for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Mar 25 2021

%K sign

%O 0,3

%A _Roger L. Bagula_, Jan 20 2009

%E Edited by _G. C. Greubel_, Mar 25 2021

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Last modified December 4 19:07 EST 2022. Contains 358563 sequences. (Running on oeis4.)