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

0,5

Links

Table of n, a(n) for n=0..51.

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.

Example

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,...].

Prog

(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))}

Crossrefs

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

Cf. related table: A122888.

Sequence in context: A060637 A123486 A350161 * A274106 A354802 A158982

Adjacent sequences: A158261 A158262 A158263 * A158265 A158266 A158267

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 16 2009

Status

approved