0,5

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.

Table of coefficients in the (2^n)-th iteration of x+x^2 begins:

1,1,0,0,0,0,0,0,0,0,0,0,0,0,...;

1,2,2,1,0,0,0,0,0,0,0,0,0,0,...;

1,4,12,30,64,118,188,258,302,298,244,162,84,32,8,1,0,0,0,0,0,...;

1,8,56,364,2240,13188,74760,409836,2179556,11271436,56788112,...;

1,16,240,3480,49280,685160,9383248,126855288,1695695976,...;

1,32,992,30256,912640,27297360,810903456,23950328688,...;

1,64,4032,252000,15665664,969917088,59855127360,3683654668512,...;

1,128,16256,2056384,259445760,32668147008,4106848523904,...;

1,256,65280,16613760,4222658560,1072200161920,272033712041216,...;

1,512,261632,133563136,68139438080,34745409189120,17710292513905152,...;

...

The initial column g.f.s are as follows:

k=1: 1/(1-2x);

k=2: 2x/((1-2x)(1-4x));

k=3: (x+16x^2)/((1-2x)(1-4x)(1-8x));

k=4: (64x^2+320x^3)/((1-2x)(1-4x)(1-8x)(1-16x));

k=5: (118x^2+5872x^3+13824x^4)/((1-2x)(1-4x)(1-8x)(1-16x)(1-32x));

...

The coefficients in the numerators of column g.f.s forms a triangle:

1;

0,2;

0,1,16;

0,0,64,320;

0,0,118,5872,13824;

0,0,188,51072,942592,1179648;

0,0,258,344304,28261632,278323200,179306496;

0,0,302,2025536,610203136,25398255616,152690491392,37044092928; ...

in which the main diagonal starts:

[1,2,16,320,13824,1179648,179306496,37044092928,-9947144257536,...];

and the row sums of the triangle begin:

[1,2,17,384,19814,2173500,486235890,215745068910,186016597075722,...].

(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))}

Cf. diagonals: A158260, A158261, A158262, A158263.

Cf. related table: A122888.

Sequence in context: A157028 A060637 A123486 this_sequence A158982 A127124 A127136

Adjacent sequences: A158261 A158262 A158263 this_sequence A158265 A158266 A158267

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 16 2009