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%I #34 Sep 08 2022 08:46:10
%S 1,-2,1,7,-5,1,-20,22,-8,1,61,-86,46,-11,1,-182,319,-224,79,-14,1,547,
%T -1139,991,-461,121,-17,1,-1640,3964,-4112,2374,-824,172,-20,1,4921,
%U -13532,16300,-11234,4846,-1340,232,-23,1,-14762,45517,-62432,50002,-25772,8866,-2036,301,-26,1,44287,-151313,232813,-212438,127318,-52370,14974,-2939,379,-29,1
%N Triangle read by rows: T(n,k) is the coefficient A_k in the transformation of 1 + x + x^2 + ... + x^n to the polynomial A_k*(x+3)^k for 0 <= k <= n.
%C Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+3)^0 + A_1*(x+3)^1 + A_2*(x+3)^2 + ... + A_n*(x+3)^n. This sequence gives A_0, ..., A_n as the entries in the n-th row of this triangle, starting at n = 0.
%H G. C. Greubel, <a href="/A248811/b248811.txt">Rows n=0..100 of triangle, flattened</a>
%F T(n,n-1) = -3*n + 1 for n > 0.
%F T(n,0) = A014983(n+1).
%F T(n,1) = (-1)^(n+1)*A191008(n-1).
%F Row n sums to A077925(n).
%e 1;
%e -2, 1;
%e 7, -5, 1;
%e -20, 22, -8, 1;
%e 61, -86, 46, -11, 1;
%e -182, 319, -224, 79, -14, 1;
%e 547, -1139, 991, -461, 121, -17, 1;
%e -1640, 3964, -4112, 2374, -824, 172, -20, 1;
%e 4921, -13532, 16300, -11234, 4846, -1340, 232, -23, 1;
%e -14762, 45517, -62432, 50002, -25772, 8866, -2036, 301, -26, 1;
%e 44287, -151313, 232813, -212438, 127318, -52370, 14974, -2939, 379, -29, 1;
%t T[n_, k_]:= Sum[(-3)^(j-k)*Binomial[j,k], {j,0,n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, May 27 2018 *)
%o (PARI) for(n=0,20,for(k=0,n,print1(sum(i=0,n,((-3)^(i-k)* binomial(i, k)) ),", ")))
%o (Magma) [[(&+[(-3)^(j-k)*Binomial(j,k): j in [0..n]]): k in [0..n]]: n in [0..20]]; // _G. C. Greubel_, May 27 2018
%Y Cf. A193843, A077925, A191008, A014983.
%K sign,tabl
%O 0,2
%A _Derek Orr_, Oct 14 2014
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