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%I #4 Sep 13 2013 19:04:35
%S 1,2,1,9,4,1,78,30,6,1,1038,364,63,8,1,18968,6233,986,108,10,1,443595,
%T 139008,20685,2072,165,12,1,12681960,3833052,545736,51494,3750,234,14,
%U 1,429244197,126105168,17365336,1569920,107760,6148,315,16,1,16801151910
%N Triangle, read by rows, that transforms diagonals in the table of coefficients of successive iterations of x*(1+x)^2 (cf. A166888).
%e Triangle begins:
%e 1;
%e 2,1;
%e 9,4,1;
%e 78,30,6,1;
%e 1038,364,63,8,1;
%e 18968,6233,986,108,10,1;
%e 443595,139008,20685,2072,165,12,1;
%e 12681960,3833052,545736,51494,3750,234,14,1;
%e 429244197,126105168,17365336,1569920,107760,6148,315,16,1;
%e 16801151910,4824243516,647216568,56661004,3728952,200583,9394,408,18,1;
%e 746998729887,210489178476,27653205177,2361036896,150566205,7768320,343063,13616,513,20,1;
%e 37200237947376,10318212622770,1332422277828,111501524409,6938694600,347030328,14703080,550300,18942,630,22,1; ...
%e Coefficients in iterations of x*(1+x)^2 form table A166888:
%e 1;
%e 1,2,1;
%e 1,4,10,18,23,22,15,6,1;
%e 1,6,27,102,333,960,2472,5748,12150,23388,40926,64872,92772,...;
%e 1,8,52,300,1578,7692,35094,150978,615939,2393628,8892054,...;
%e 1,10,85,660,4790,32920,215988,1360638,8265613,48585702,...;
%e 1,12,126,1230,11385,101010,864813,7178700,57976074,456783888,...;
%e 1,14,175,2058,23163,251832,2660028,27405798,276215313,...;
%e 1,16,232,3192,42308,544600,6842220,84191772,1017153322,...;
%e ...
%e This triangle T transforms one diagonal in A166888 into another,
%e for example: T * A154256 = A119820, T * A119820 = A166889, where
%e A154256 = [1,2,10,102,1578,32920,864813,27405798,1017153322,...];
%e A119820 = [1,4,27,300,4790,101010,2660028,84191772,3115739358,...];
%e A166889 = [1,6,52,660,11385,251832,6842220,221228244,8311401351,...].
%o (PARI) {T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+2*x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
%Y Cf. columns: A166891, A166892, A166893; A229113 (row sums).
%Y Cf. variants: A135080, A166884.
%Y Cf. A166888, A154256, A119820, A166889.
%K nonn,tabl
%O 1,2
%A _Paul D. Hanna_, Nov 22 2009
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