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A166888 Triangle T(n,k), read by rows n>=0 with terms k=1..3^n, where row n lists the coefficients in the n-th iteration of x*(1+x)^2.

%I A166888 #2 Mar 30 2012 18:37:18
%S A166888 1,1,2,1,1,4,10,18,23,22,15,6,1,1,6,27,102,333,960,2472,5748,12150,
%T A166888 23388,40926,64872,92772,119216,137112,140526,127677,102150,71331,
%U A166888 42954,21939,9288,3156,822,153,18,1,1,8,52,300,1578,7692,35094,150978
%N A166888 Triangle T(n,k), read by rows n>=0 with terms k=1..3^n, where row n lists the coefficients in the n-th iteration of x*(1+x)^2.
%e A166888 Triangle begins:
%e A166888 1;
%e A166888 1,2,1;
%e A166888 1,4,10,18,23,22,15,6,1;
%e A166888 1,6,27,102,333,960,2472,5748,12150,23388,40926,64872,92772,...;
%e A166888 1,8,52,300,1578,7692,35094,150978,615939,2393628,8892054,...;
%e A166888 1,10,85,660,4790,32920,215988,1360638,8265613,48585702,...;
%e A166888 1,12,126,1230,11385,101010,864813,7178700,57976074,456783888,...;
%e A166888 1,14,175,2058,23163,251832,2660028,27405798,276215313,...;
%e A166888 1,16,232,3192,42308,544600,6842220,84191772,1017153322,...;
%e A166888 1,18,297,4680,71388,1061712,15463512,221228244,3115739358,...;
%e A166888 1,20,370,6570,113355,1912590,31683051,516686346,8311401351,...;
%e A166888 1,22,451,8910,171545,3237520,60108576,1100544720,19906483168,...;
%e A166888 1,24,540,11748,249678,5211492,107184066,2176952910,43733857365,...;
%e A166888 ...
%e A166888 The initial diagonals in this triangle begin:
%e A166888 A154256 = [1,2,10,102,1578,32920,864813,27405798,1017153322,...];
%e A166888 A119820 = [1,4,27,300,4790,101010,2660028,84191772,3115739358,...];
%e A166888 A166889 = [1,6,52,660,11385,251832,6842220,221228244,8311401351,...].
%e A166888 The diagonals are transformed one into the other by
%e A166888 triangle A166890, which begins:
%e A166888 1;
%e A166888 2,1;
%e A166888 9,4,1;
%e A166888 78,30,6,1;
%e A166888 1038,364,63,8,1;
%e A166888 18968,6233,986,108,10,1;
%e A166888 443595,139008,20685,2072,165,12,1;
%e A166888 12681960,3833052,545736,51494,3750,234,14,1; ...
%o A166888 (PARI) {T(n, k)=local(F=x+2*x^2+x^3, G=x+x*O(x^k)); if(n<0, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, k, x)))}
%Y A166888 Cf. diagonals: A154256, A119820, A166889, variants: A166880, A122888.
%K A166888 nonn,tabf
%O A166888 0,3
%A A166888 _Paul D. Hanna_, Nov 22 2009