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A248826 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+k)^k for 0 <= k <= n.

%I A248826 #29 Oct 23 2014 20:49:44
%S A248826 1,0,1,0,-3,1,0,6,-8,1,0,-10,40,-15,1,0,15,-160,135,-24,1,0,-21,560,
%T A248826 -945,336,-35,1,0,28,-1792,5670,-3584,700,-48,1,0,-36,5376,-30618,
%U A248826 32256,-10500,1296,-63,1,0,45,-15360,153090,-258048,131250,-25920,2205,-80,1,0,-55,42240,-721710,1892352,-1443750,427680,-56595,3520,-99,1
%N A248826 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+k)^k for 0 <= k <= n.
%C A248826 Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x+0)^0 + A_1*(x+1)^1 + A_2*(x+2)^2 + ... + A_n*(x+n)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.
%F A248826 T(n,1) = n*(n+1)*(-1)^(n+1)/2 for n > 0.
%F A248826 T(n,2) = Binomial(n+1,3)*2^(n-2)*(-1)^n for n > 1.
%F A248826 T(n,n-1) = 1 - n^2 for n > 0.
%F A248826 T(n,n-2) = (1/2)*n*(n-2)^2*(n+1) for n > 1.
%e A248826 1;
%e A248826 0,   1;
%e A248826 0,  -3,      1;
%e A248826 0,   6,     -8,       1;
%e A248826 0, -10,     40,     -15,       1;
%e A248826 0,  15,   -160,     135,     -24,        1;
%e A248826 0, -21,    560,    -945,     336,      -35,      1;
%e A248826 0,  28,  -1792,    5670,   -3584,      700,    -48,      1;
%e A248826 0, -36,   5376,  -30618,   32256,   -10500,   1296,    -63,    1;
%e A248826 0,  45, -15360,  153090, -258048,   131250, -25920,   2205,  -80,   1;
%e A248826 0, -55,  42240, -721710, 1892352, -1443750, 427680, -56595, 3520, -99, 1;
%o A248826 (PARI) for(n=0,20,for(k=0,n,if(!k,if(n,print1(0,", "));if(!n,print1(1,", ")));if(k,print1(-sum(i=1,n,((-k)^(i-k-1)*i*binomial(i,k))),", "))))
%Y A248826 Cf. A001789, A000217.
%K A248826 sign,tabl
%O A248826 0,5
%A A248826 _Derek Orr_, Oct 15 2014