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.

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, 2240, 118, 1, 128, 4032, 30256, 49280, 13188, 188, 1, 256, 16256, 252000, 912640, 685160, 74760, 258, 1, 512, 65280, 2056384, 15665664, 27297360, 9383248, 409836, 302, 1

Offset: 0

Author

Paul D. Hanna, Mar 16 2009

Keywords

• nonn
• tabl

Examples

			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,...].
		

Crossrefs

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

Cf. related table: A122888.

Programs

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))}

Formula

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.