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%I #19 Oct 23 2014 20:47:35
%S 1,3,1,3,9,1,3,45,19,1,3,173,211,33,1,3,573,1811,633,51,1,3,1725,
%T 13331,9273,1491,73,1,3,4861,88595,115113,32851,3013,99,1,3,13053,
%U 547347,1276329,606291,92613,5475,129,1,3,33789,3201555,13033641,9896019,2360613,223203,9201,163,1,3,84989,17947155,125008041,147521619,52760613,7480803,479601,14563,201,1
%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-2k)^k for 0 <= k <= n.
%C Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-0)^0 + A_1*(x-2)^1 + A_2*(x-4)^2 + ... + A_n*(x-2n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.
%F T(n,n-1) = 2*n^2 + 1 for n > 0.
%F T(n,1) = 2^n*(n^2-2*n+3)-3 for n > 0.
%e 1;
%e 3, 1;
%e 3, 9, 1;
%e 3, 45, 19, 1;
%e 3, 173, 211, 33, 1;
%e 3, 573, 1811, 633, 51, 1;
%e 3, 1725, 13331, 9273, 1491, 73, 1;
%e 3, 4861, 88595, 115113, 32851, 3013, 99, 1;
%e 3, 13053, 547347, 1276329, 606291, 92613, 5475, 129, 1;
%e 3, 33789, 3201555, 13033641, 9896019, 2360613, 223203, 9201, 163, 1;
%o (PARI) for(n=0,10,for(k=0,n,if(!k,if(n,print1(3,", "));if(!n,print1(1,", ")));if(k,print1(sum(i=1,n,((2*k)^(i-k)*i*binomial(i,k)))/k,", "))))
%Y Cf. A036826, A058331, A248829, A242598.
%K nonn,tabl
%O 0,2
%A _Derek Orr_, Oct 15 2014
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