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A166880 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+x^2+x^3).

%I A166880 #2 Mar 30 2012 18:37:18
%S A166880 1,1,1,1,1,2,4,6,8,8,6,3,1,1,3,9,24,60,138,294,579,1053,1767,2739,
%T A166880 3924,5196,6352,7152,7389,6969,5961,4587,3144,1896,990,438,159,45,9,1,
%U A166880 1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,3730846,9262712
%N A166880 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+x^2+x^3).
%e A166880 Triangle begins:
%e A166880 1;
%e A166880 1,1,1;
%e A166880 1,2,4,6,8,8,6,3,1;
%e A166880 1,3,9,24,60,138,294,579,1053,1767,2739,3924,5196,6352,7152,7389,6969,5961,4587,3144,1896,990,438,159,45,9,1;
%e A166880 1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,...;
%e A166880 1,5,25,120,560,2540,11220,48330,203230,835080,3355950,13200648,...;
%e A166880 1,6,36,210,1200,6720,36930,199365,1058175,5526330,28417200,...;
%e A166880 1,7,49,336,2268,15078,98826,639093,4080531,25738755,160474545,...;
%e A166880 1,8,64,504,3920,30128,228984,1722084,12821788,94556532,...;
%e A166880 1,9,81,720,6336,55224,477000,4085028,34700940,292495896,...;
%e A166880 1,10,100,990,9720,94680,915390,8787735,83795085,793894860,...;
%e A166880 1,11,121,1320,14300,153890,1645710,17494455,184915225,...;
%e A166880 1,12,144,1716,20328,239448,2805396,32700558,379309986,...;
%e A166880 1,13,169,2184,28080,359268,4575324,58009614,732380298,...;
%e A166880 1,14,196,2730,37856,522704,7188090,98465913,1343828395,...;
%e A166880 1,15,225,3360,49980,740670,10937010,160947465,2360704815,...;
%e A166880 1,16,256,4080,64800,1025760,16185840,254624520,3993857400,...;
%e A166880 1,17,289,4896,82688,1392368,23379216,391488648,6538326616,...;
%e A166880 1,18,324,5814,104040,1856808,33053814,586957419,10398271833,...;
%e A166880 ...
%e A166880 The initial diagonals in this triangle begin:
%e A166880 A166881: [1,1,4,24,216,2540,36930,639093,12821788,292495896,...];
%e A166880 A166882: [1,2,9,60,560,6720,98826,1722084,34700940,793894860,...];
%e A166880 A166883: [1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,...]; ...
%e A166880 The diagonals are transformed one into the other by
%e A166880 triangle A166884, which begins:
%e A166880 1;
%e A166880 1,1;
%e A166880 3,2,1;
%e A166880 15,9,3,1;
%e A166880 114,62,18,4,1;
%e A166880 1159,593,157,30,5,1;
%e A166880 14838,7266,1812,316,45,6,1;
%e A166880 229401,108720,25989,4271,555,63,7,1;
%e A166880 4159662,1922166,445255,70180,8595,890,84,8,1; ...
%o A166880 (PARI) {T(n, k)=local(F=x+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 A166880 Cf. diagonals: A166881, A166882, A166883, related triangle: A166884.
%Y A166880 Cf. row sums: A166999, variant: A122888.
%K A166880 nonn,tabf
%O A166880 0,6
%A A166880 _Paul D. Hanna_, Nov 21 2009