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%I #8 Jan 26 2013 19:20:56
%S 1,0,1,0,1,1,0,0,2,1,0,0,3,3,1,0,0,2,6,4,1,0,0,1,10,10,5,1,0,0,0,12,
%T 20,15,6,1,0,0,0,12,35,35,21,7,1,0,0,0,10,52,70,56,28,8,1,0,0,0,6,68,
%U 126,126,84,36,9,1,0,0,0,3,80,205,252,210,120,45,10,1
%N Triangle, read by rows, where T(n,k) = [x^n] x^k*(1-x^k)^(k-1) / (1-x)^(k-1) for n>=k>=1.
%F G.f. of column k = x^k * ( (1-x^k)/(1-x) )^(k-1).
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 0, 2, 1;
%e 0, 0, 3, 3, 1;
%e 0, 0, 2, 6, 4, 1;
%e 0, 0, 1, 10, 10, 5, 1;
%e 0, 0, 0, 12, 20, 15, 6, 1;
%e 0, 0, 0, 12, 35, 35, 21, 7, 1;
%e 0, 0, 0, 10, 52, 70, 56, 28, 8, 1;
%e 0, 0, 0, 6, 68, 126, 126, 84, 36, 9, 1;
%e 0, 0, 0, 3, 80, 205, 252, 210, 120, 45, 10, 1;
%e 0, 0, 0, 1, 85, 305, 462, 462, 330, 165, 55, 11, 1;
%e 0, 0, 0, 0, 80, 420, 786, 924, 792, 495, 220, 66, 12, 1; ...
%e in which column sums equal: A000169(k) = k^(k-1) for k>=1.
%o (PARI) {T(n,k)=polcoeff(((1-x^k)/(1-x +x*O(x^n)))^(k-1),n-k)}
%o for(n=1,12,for(k=1,n,print1(T(n,k),", "));print(""))
%Y Cf. A221834, A000169, A077227.
%K nonn,tabl
%O 1,9
%A _Paul D. Hanna_, Jan 26 2013
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