A166880 - OEIS

%I #2 Mar 30 2012 18:37:18

%S 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 3924,5196,6352,7152,7389,6969,5961,4587,3144,1896,990,438,159,45,9,1,

%U 1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,3730846,9262712

%N 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 Triangle begins:

%e 1;

%e 1,1,1;

%e 1,2,4,6,8,8,6,3,1;

%e 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 1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,...;

%e 1,5,25,120,560,2540,11220,48330,203230,835080,3355950,13200648,...;

%e 1,6,36,210,1200,6720,36930,199365,1058175,5526330,28417200,...;

%e 1,7,49,336,2268,15078,98826,639093,4080531,25738755,160474545,...;

%e 1,8,64,504,3920,30128,228984,1722084,12821788,94556532,...;

%e 1,9,81,720,6336,55224,477000,4085028,34700940,292495896,...;

%e 1,10,100,990,9720,94680,915390,8787735,83795085,793894860,...;

%e 1,11,121,1320,14300,153890,1645710,17494455,184915225,...;

%e 1,12,144,1716,20328,239448,2805396,32700558,379309986,...;

%e 1,13,169,2184,28080,359268,4575324,58009614,732380298,...;

%e 1,14,196,2730,37856,522704,7188090,98465913,1343828395,...;

%e 1,15,225,3360,49980,740670,10937010,160947465,2360704815,...;

%e 1,16,256,4080,64800,1025760,16185840,254624520,3993857400,...;

%e 1,17,289,4896,82688,1392368,23379216,391488648,6538326616,...;

%e 1,18,324,5814,104040,1856808,33053814,586957419,10398271833,...;

%e ...

%e The initial diagonals in this triangle begin:

%e A166881: [1,1,4,24,216,2540,36930,639093,12821788,292495896,...];

%e A166882: [1,2,9,60,560,6720,98826,1722084,34700940,793894860,...];

%e A166883: [1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,...]; ...

%e The diagonals are transformed one into the other by

%e triangle A166884, which begins:

%e 1;

%e 1,1;

%e 3,2,1;

%e 15,9,3,1;

%e 114,62,18,4,1;

%e 1159,593,157,30,5,1;

%e 14838,7266,1812,316,45,6,1;

%e 229401,108720,25989,4271,555,63,7,1;

%e 4159662,1922166,445255,70180,8595,890,84,8,1; ...

%o (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 Cf. diagonals: A166881, A166882, A166883, related triangle: A166884.

%Y Cf. row sums: A166999, variant: A122888.

%K nonn,tabf

%O 0,6

%A _Paul D. Hanna_, Nov 21 2009