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 1-3 of 3 results.

A166888 Triangle T(n,k), read by rows n>=0 with terms k=1..3^n, where row n lists the coefficients in the n-th iteration of x*(1+x)^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 10, 18, 23, 22, 15, 6, 1, 1, 6, 27, 102, 333, 960, 2472, 5748, 12150, 23388, 40926, 64872, 92772, 119216, 137112, 140526, 127677, 102150, 71331, 42954, 21939, 9288, 3156, 822, 153, 18, 1, 1, 8, 52, 300, 1578, 7692, 35094, 150978
Offset: 0

Views

Author

Paul D. Hanna, Nov 22 2009

Keywords

Examples

			Triangle begins:
1;
1,2,1;
1,4,10,18,23,22,15,6,1;
1,6,27,102,333,960,2472,5748,12150,23388,40926,64872,92772,...;
1,8,52,300,1578,7692,35094,150978,615939,2393628,8892054,...;
1,10,85,660,4790,32920,215988,1360638,8265613,48585702,...;
1,12,126,1230,11385,101010,864813,7178700,57976074,456783888,...;
1,14,175,2058,23163,251832,2660028,27405798,276215313,...;
1,16,232,3192,42308,544600,6842220,84191772,1017153322,...;
1,18,297,4680,71388,1061712,15463512,221228244,3115739358,...;
1,20,370,6570,113355,1912590,31683051,516686346,8311401351,...;
1,22,451,8910,171545,3237520,60108576,1100544720,19906483168,...;
1,24,540,11748,249678,5211492,107184066,2176952910,43733857365,...;
...
The initial diagonals in this triangle begin:
A154256 = [1,2,10,102,1578,32920,864813,27405798,1017153322,...];
A119820 = [1,4,27,300,4790,101010,2660028,84191772,3115739358,...];
A166889 = [1,6,52,660,11385,251832,6842220,221228244,8311401351,...].
The diagonals are transformed one into the other by
triangle A166890, which begins:
1;
2,1;
9,4,1;
78,30,6,1;
1038,364,63,8,1;
18968,6233,986,108,10,1;
443595,139008,20685,2072,165,12,1;
12681960,3833052,545736,51494,3750,234,14,1; ...

Crossrefs

Cf. diagonals: A154256, A119820, A166889, variants: A166880, A122888.

Programs

  • PARI
    {T(n, k)=local(F=x+2*x^2+x^3, G=x+x*O(x^k)); if(n<0, 0, for(i=1, n, G=subst(F, x, G)); return(polcoeff(G, k, x)))}

A166890 Triangle, read by rows, that transforms diagonals in the table of coefficients of successive iterations of x*(1+x)^2 (cf. A166888).

Original entry on oeis.org

1, 2, 1, 9, 4, 1, 78, 30, 6, 1, 1038, 364, 63, 8, 1, 18968, 6233, 986, 108, 10, 1, 443595, 139008, 20685, 2072, 165, 12, 1, 12681960, 3833052, 545736, 51494, 3750, 234, 14, 1, 429244197, 126105168, 17365336, 1569920, 107760, 6148, 315, 16, 1, 16801151910
Offset: 1

Views

Author

Paul D. Hanna, Nov 22 2009

Keywords

Examples

			Triangle begins:
1;
2,1;
9,4,1;
78,30,6,1;
1038,364,63,8,1;
18968,6233,986,108,10,1;
443595,139008,20685,2072,165,12,1;
12681960,3833052,545736,51494,3750,234,14,1;
429244197,126105168,17365336,1569920,107760,6148,315,16,1;
16801151910,4824243516,647216568,56661004,3728952,200583,9394,408,18,1;
746998729887,210489178476,27653205177,2361036896,150566205,7768320,343063,13616,513,20,1;
37200237947376,10318212622770,1332422277828,111501524409,6938694600,347030328,14703080,550300,18942,630,22,1; ...
Coefficients in iterations of x*(1+x)^2 form table A166888:
1;
1,2,1;
1,4,10,18,23,22,15,6,1;
1,6,27,102,333,960,2472,5748,12150,23388,40926,64872,92772,...;
1,8,52,300,1578,7692,35094,150978,615939,2393628,8892054,...;
1,10,85,660,4790,32920,215988,1360638,8265613,48585702,...;
1,12,126,1230,11385,101010,864813,7178700,57976074,456783888,...;
1,14,175,2058,23163,251832,2660028,27405798,276215313,...;
1,16,232,3192,42308,544600,6842220,84191772,1017153322,...;
...
This triangle T transforms one diagonal in A166888 into another,
for example: T * A154256 = A119820, T * A119820 = A166889, where
A154256 = [1,2,10,102,1578,32920,864813,27405798,1017153322,...];
A119820 = [1,4,27,300,4790,101010,2660028,84191772,3115739358,...];
A166889 = [1,6,52,660,11385,251832,6842220,221228244,8311401351,...].

Crossrefs

Cf. columns: A166891, A166892, A166893; A229113 (row sums).
Cf. variants: A135080, A166884.
Cf. A166888, A154256, A119820, A166889.

Programs

  • PARI
    {T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+2*x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}

A166889 Coefficients of x^n in the (n+1)-th iteration of x*(1+x)^2 for n>=1.

Original entry on oeis.org

1, 6, 52, 660, 11385, 251832, 6842220, 221228244, 8311401351, 356190316416, 17160064580802, 918453056609946, 54085054802995008, 3475794779752572784, 242103490865991893116, 18170143514998451547348
Offset: 1

Views

Author

Paul D. Hanna, Nov 22 2009

Keywords

Examples

			Coefficients in the initial iterations of x*(1+x)^2 begin:
[1,2,1];
[(1),4,10,18,23,22,15,6,1];
[1,(6),27,102,333,960,2472,5748,12150,23388,...];
[1,8,(52),300,1578,7692,35094,150978,615939,2393628,...];
[1,10,85,(660),4790,32920,215988,1360638,8265613,48585702,...];
[1,12,126,1230,(11385),101010,864813,7178700,57976074,...];
[1,14,175,2058,23163,(251832),2660028,27405798,276215313,...];
[1,16,232,3192,42308,544600,(6842220),84191772,1017153322,...];
[1,18,297,4680,71388,1061712,15463512,(221228244),3115739358,...];
[1,20,370,6570,113355,1912590,31683051,516686346,(8311401351),...]; ...
where the coefficients in parenthesis form the initial terms of this sequence.

Crossrefs

Cf. A166888, A154256, A119820, A166890.

Programs

  • PARI
    {a(n)=local(F=x*(1+x)^2, G=x+x*O(x^n)); if(n<1, 0, for(i=1, n+1, G=subst(F, x, G)); return(polcoeff(G, n, x)))}
Showing 1-3 of 3 results.

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