cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A158261 a(n) = coefficient of x^n in the (2^(n-1))-th iteration of x+x^2 for n>=1.

Original entry on oeis.org

1, 2, 12, 364, 49280, 27297360, 59855127360, 515600292989376, 17478262497392546432, 2341170788069821884925696, 1243964516830446590036473921536, 2629751195406987463208250109864126464
Offset: 1

Views

Author

Paul D. Hanna, Mar 15 2009

Keywords

Examples

			Table of coefficients in the (2^i)-th iteration of x+x^2 begins:
(1),1;
1,(2),2,1;
1,4,(12),30,64,118,188,258,302,298,244,162,84,32,8,1;
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,(515600292989376),...;
1,256,65280,16613760,4222658560,1072200161920,272033712041216,68973668731925376,(17478262497392546432),...;
...
where the terms in parenthesis form the initial terms of this sequence.

Links

Crossrefs

Cf. A158260, A158262, A158263, A158264 (table).

Programs

  • PARI
    {a(n)=local(G=x+x^2+x*O(x^n)); if(n<0, 0, for(i=1, n-1, G=subst(G, x, G+x*O(x^n))); polcoeff(G, n, x))}

A158262 a(n) = coefficient of x^n in the (2^n)-th iteration of x+x^2 for n>=1.

Original entry on oeis.org

1, 4, 56, 3480, 912640, 969917088, 4106848523904, 68973668731925376, 4597014244761562326272, 1218175506582318206655794688, 1285897546575275148015361075150848
Offset: 1

Views

Author

Paul D. Hanna, Mar 15 2009

Keywords

Examples

			Table of coefficients in the (2^i)-th iteration of x+x^2 begins:
1,1;
(1),2,2,1;
1,(4),12,30,64,118,188,258,302,298,244,162,84,32,8,1;
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,(68973668731925376),...;
...
Where the terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A158260, A158261, A158263, A158264 (table).

Programs

  • PARI
    {a(n)=local(G=x+x^2+x*O(x^n)); if(n<0, 0, for(i=1, n, G=subst(G, x, G+x*O(x^n))); polcoeff(G, n, x))}

A158263 a(n) = coefficient of x^n in the (2^(n+1))-th iteration of x+x^2 for n>=1.

Original entry on oeis.org

1, 8, 240, 30256, 15665664, 32668147008, 272033712041216, 9024264001164470016, 1192791193150627685091840, 628748300357129103400036998144, 1322980853407936018020929177243811840
Offset: 1

Views

Author

Paul D. Hanna, Mar 15 2009

Keywords

Examples

			Table of coefficients in the (2^i)-th iteration of x+x^2 begins:
1,1;
1,2,2,1;
(1),4,12,30,64,118,188,258,302,298,244,162,84,32,8,1;
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,(9024264001164470016),...;
...
Where the terms in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A158260, A158261, A158262, A158264 (table).

Programs

  • PARI
    {a(n)=local(G=x+x^2+x*O(x^n)); if(n<0, 0, for(i=1, n+1, G=subst(G, x, G+x*O(x^n))); polcoeff(G, n, x))}

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.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Mar 16 2009

Keywords

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.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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