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A158264 Table where row n lists the coefficients in the (2^n)-th iteration of x+x^2 for n>=0, read by antidiagonals not including trailing zeros in rows.

%I A158264 #2 Mar 30 2012 18:37:16
%S A158264 1,1,1,1,2,1,4,2,1,8,12,1,1,16,56,30,1,32,240,364,64,1,64,992,3480,
%T A158264 2240,118,1,128,4032,30256,49280,13188,188,1,256,16256,252000,912640,
%U A158264 685160,74760,258,1,512,65280,2056384,15665664,27297360,9383248,409836,302,1
%N A158264 Table where row n lists the coefficients in the (2^n)-th iteration of x+x^2 for n>=0, read by antidiagonals not including trailing zeros in rows.
%F A158264 G.f. of column k: P_k(x)/Product_{j=1,k} (1-2^j*x) where P_k(x) is a polynomial of degree k-1 for k>=1.
%e A158264 Table of coefficients in the (2^n)-th iteration of x+x^2 begins:
%e A158264 1,1,0,0,0,0,0,0,0,0,0,0,0,0,...;
%e A158264 1,2,2,1,0,0,0,0,0,0,0,0,0,0,...;
%e A158264 1,4,12,30,64,118,188,258,302,298,244,162,84,32,8,1,0,0,0,0,0,...;
%e A158264 1,8,56,364,2240,13188,74760,409836,2179556,11271436,56788112,...;
%e A158264 1,16,240,3480,49280,685160,9383248,126855288,1695695976,...;
%e A158264 1,32,992,30256,912640,27297360,810903456,23950328688,...;
%e A158264 1,64,4032,252000,15665664,969917088,59855127360,3683654668512,...;
%e A158264 1,128,16256,2056384,259445760,32668147008,4106848523904,...;
%e A158264 1,256,65280,16613760,4222658560,1072200161920,272033712041216,...;
%e A158264 1,512,261632,133563136,68139438080,34745409189120,17710292513905152,...;
%e A158264 ...
%e A158264 The initial column g.f.s are as follows:
%e A158264 k=1: 1/(1-2x);
%e A158264 k=2: 2x/((1-2x)(1-4x));
%e A158264 k=3: (x+16x^2)/((1-2x)(1-4x)(1-8x));
%e A158264 k=4: (64x^2+320x^3)/((1-2x)(1-4x)(1-8x)(1-16x));
%e A158264 k=5: (118x^2+5872x^3+13824x^4)/((1-2x)(1-4x)(1-8x)(1-16x)(1-32x));
%e A158264 ...
%e A158264 The coefficients in the numerators of column g.f.s forms a triangle:
%e A158264 1;
%e A158264 0,2;
%e A158264 0,1,16;
%e A158264 0,0,64,320;
%e A158264 0,0,118,5872,13824;
%e A158264 0,0,188,51072,942592,1179648;
%e A158264 0,0,258,344304,28261632,278323200,179306496;
%e A158264 0,0,302,2025536,610203136,25398255616,152690491392,37044092928; ...
%e A158264 in which the main diagonal starts:
%e A158264 [1,2,16,320,13824,1179648,179306496,37044092928,-9947144257536,...];
%e A158264 and the row sums of the triangle begin:
%e A158264 [1,2,17,384,19814,2173500,486235890,215745068910,186016597075722,...].
%o A158264 (PARI) {T(n, k)=local(G=x+x^2+x*O(x^k)); if(n<1, 0,for(i=1, n-1, G=subst(G, x, G)); polcoeff(G, k, x))}
%Y A158264 Cf. diagonals: A158260, A158261, A158262, A158263.
%Y A158264 Cf. related table: A122888.
%K A158264 nonn,tabl
%O A158264 0,5
%A A158264 _Paul D. Hanna_, Mar 16 2009