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 1-6 of 6 results.

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]}

A166891 Column 1 of triangle A166890.

Original entry on oeis.org

1, 2, 9, 78, 1038, 18968, 443595, 12681960, 429244197, 16801151910, 746998729887, 37200237947376, 2051666003699226, 124156748403386646, 8180285024067867345, 582970677419310149580, 44684461723038752605932
Offset: 1

Views

Author

Paul D. Hanna, Nov 22 2009

Keywords

Comments

Triangle A166890 transforms diagonals in the triangle A166888 of coefficients in the successive iterations of x*(1+x)^2.

Crossrefs

Cf. A166890, A166892, A166893.

Programs

  • PARI
    {a(n)=local(F=x, M, N, P, m=n); 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]}

A166892 Column 2 of triangle A166890.

Original entry on oeis.org

1, 4, 30, 364, 6233, 139008, 3833052, 126105168, 4824243516, 210489178476, 10318212622770, 561491367744672, 33588989930164050, 2190978413703916624, 154771816676784778818, 11771103512077651149912, 959000166676677798631894
Offset: 1

Views

Author

Paul D. Hanna, Nov 22 2009

Keywords

Comments

Triangle A166890 transforms diagonals in the triangle A166888 of coefficients in the successive iterations of x*(1+x)^2.

Crossrefs

Cf. A166890, A166891, A166893.

Programs

  • PARI
    {a(n)=local(F=x, M, N, P, m=n+1); 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, 2]}

A166893 Column 3 of triangle A166890.

Original entry on oeis.org

1, 6, 63, 986, 20685, 545736, 17365336, 647216568, 27653205177, 1332422277828, 71470510481961, 4223498675806638, 272615162534575302, 19082609490868539738, 1439738711122827542742, 116468234559061615308870
Offset: 1

Views

Author

Paul D. Hanna, Nov 22 2009

Keywords

Comments

Triangle A166890 transforms diagonals in the triangle A166888 of coefficients in the successive iterations of x*(1+x)^2.

Crossrefs

Cf. A166890, A166891, A166892.

Programs

  • PARI
    {a(n)=local(F=x, M, N, P, m=n+2); 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+2, 3]}

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)))}

A229113 Row sums of triangle A166890.

Original entry on oeis.org

1, 3, 14, 115, 1474, 26306, 605538, 17116241, 574398861, 22333212354, 987660842174, 48969675370286, 2690888441513773, 162333780684342177, 10667116965172284025, 758434434866124781819, 58015602766701416940599, 4750972163883049286678765, 414721061026426313701858479
Offset: 0

Views

Author

Paul D. Hanna, Sep 13 2013

Keywords

Comments

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

Examples

			Triangle A166890 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; ...
of which the row sums form this sequence.

Crossrefs

Cf. A166890.

Programs

  • PARI
    {a(n, k)=local(F=x, M, N, P, m=max(n, k), A166890); 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]); A166890=P~*(N~)^-1;sum(k=0,n,A166890[n+1, k+1])}
for(n=0, 25, print1(a(n), ", "))
Showing 1-6 of 6 results.

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