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 1-10 of 10 results.

A166884 Triangle, read by rows, that transforms diagonals in the table of coefficients of successive iterations of x+x^2+x^3 (cf. A166880).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 15, 9, 3, 1, 114, 62, 18, 4, 1, 1159, 593, 157, 30, 5, 1, 14838, 7266, 1812, 316, 45, 6, 1, 229401, 108720, 25989, 4271, 555, 63, 7, 1, 4159662, 1922166, 445255, 70180, 8595, 890, 84, 8, 1, 86580636, 39212154, 8865333, 1354750, 159171, 15534, 1337, 108, 9, 1
Offset: 0

Views

Author

Paul D. Hanna, Nov 21 2009

Keywords

Examples

			This triangle begins:
1;
1, 1;
3, 2, 1;
15, 9, 3, 1;
114, 62, 18, 4, 1;
1159, 593, 157, 30, 5, 1;
14838, 7266, 1812, 316, 45, 6, 1;
229401, 108720, 25989, 4271, 555, 63, 7, 1;
4159662, 1922166, 445255, 70180, 8595, 890, 84, 8, 1;
86580636, 39212154, 8865333, 1354750, 159171, 15534, 1337, 108, 9, 1;
2034850425, 906623004, 201058614, 30000676, 3418245, 320070, 25963, 1912, 135, 10, 1;
53303009286, 23429034168, 5114874693, 748896765, 83336385, 7568355, 589057, 40882, 2631, 165, 11, 1; ...
Triangle A166880 of coefficients in iterations of x+x^2+x^3 begins:
1;
1,1,1;
1,2,4,6,8,8,6,3,1;
1,3,9,24,60,138,294,579,1053,1767,2739,3924,5196,6352,7152,7389,...;
1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,...;
1,5,25,120,560,2540,11220,48330,203230,835080,3355950,13200648,...;
1,6,36,210,1200,6720,36930,199365,1058175,5526330,28417200,...;
1,7,49,336,2268,15078,98826,639093,4080531,25738755,160474545,...;
1,8,64,504,3920,30128,228984,1722084,12821788,94556532,...; ...
in which this triangle transforms diagonals in A166880 into each other.
The initial diagonals in triangle A166880 begin:
A166881: [1,1,4,24,216,2540,36930,639093,12821788,292495896,...];
A166882: [1,2,9,60,560,6720,98826,1722084,34700940,793894860,...];
A166883: [1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,...]; ...
so that, if we treat the diagonals as column vectors, we have:
A166884 * A166881 = A166882,
A166884 * A166882 = A166883.

Links

Crossrefs

Cf. A166880, columns: A166885, A166886, A166887; A229112 (row sums).

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+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]}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

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)))}

A166881 a(n) = coefficient of x^n in the (n-1)-th iteration of (x + x^2 + x^3) for n>=1.

Original entry on oeis.org

1, 1, 4, 24, 216, 2540, 36930, 639093, 12821788, 292495896, 7475306400, 211531253076, 6564750305124, 221684308001728, 8091749562745576, 317454163281499140, 13320693233434444092, 595287890670560958740, 28226111104873887744528, 1415312988632326542765024
Offset: 1

Views

Author

Paul D. Hanna, Oct 22 2009

Keywords

Examples

			Let F_n(x) denote the n-th iteration of F(x) = x + x^2 + x^3;
then coefficients in the successive iterations of F(x) begin:
F_0: [(1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
F(x):[1, (1), 1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
F_2: [1, 2, (4), 6, 8, 8, 6, 3, 1, 0, 0, ...];
F_3: [1, 3, 9, (24), 60, 138, 294, 579, 1053, 1767, 2739, ...];
F_4: [1, 4, 16, 60, (216), 744, 2460, 7818, 23910, 70446, 200160, ...];
F_5: [1, 5, 25, 120, 560, (2540), 11220, 48330, 203230, 835080, ...];
F_6: [1, 6, 36, 210, 1200, 6720, (36930), 199365, 1058175, ...];
F_7: [1, 7, 49, 336, 2268, 15078, 98826, (639093), 4080531, ...];
F_8: [1, 8, 64, 504, 3920, 30128, 228984, 1722084, (12821788),...];
F_9: [1, 9, 81, 720, 6336, 55224, 477000, 4085028, 34700940, (292495896), ...]; ...
where the coefficients along the diagonal (shown above in parenthesis)
form the initial terms of this sequence.

Links

Crossrefs

Cf. A166880, A166882, A166883, A166884.

Programs

  • PARI
    {a(n)=local(F=x+x^2+x^3, 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)))}

Extensions

Duplicate a(19) removed by Andrew Howroyd, Feb 22 2018

A166882 a(n) = coefficient of x^n in the n-th iteration of (x + x^2 + x^3) for n>=1.

Original entry on oeis.org

1, 2, 9, 60, 560, 6720, 98826, 1722084, 34700940, 793894860, 20329008975, 576026191026, 17893288364952, 604630781494558, 22079861395250568, 866509034147074284, 36367487433847501620, 1625458443704631873072
Offset: 1

Views

Author

Paul D. Hanna, Oct 22 2009

Keywords

Examples

			Let F_n(x) denote the n-th iteration of F(x) = x + x^2 + x^3;
then coefficients in the successive iterations of F(x) begin:
F(x):[(1), 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
F_2: [1, (2), 4, 6, 8, 8, 6, 3, 1, 0, 0, ...];
F_3: [1, 3, (9), 24, 60, 138, 294, 579, 1053, 1767, 2739, ...];
F_4: [1, 4, 16, (60), 216, 744, 2460, 7818, 23910, 70446, 200160, ...];
F_5: [1, 5, 25, 120, (560), 2540, 11220, 48330, 203230, 835080, ...];
F_6: [1, 6, 36, 210, 1200, (6720), 36930, 199365, 1058175, ...];
F_7: [1, 7, 49, 336, 2268, 15078, (98826), 639093, 4080531, ...];
F_8: [1, 8, 64, 504, 3920, 30128, 228984, (1722084), 12821788, ...];
F_9: [1, 9, 81, 720, 6336, 55224, 477000, 4085028, (34700940), ...];
F_10:[1, 10, 100, 990, 9720, 94680, 915390, 8787735, 83795085, (793894860), ...]; ...
where the coefficients along the diagonal (shown above in parenthesis)
form the initial terms of this sequence.

Crossrefs

Cf. A166880, A166881, A166883, A166884.

Programs

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

A166883 a(n) = coefficient of x^n in the (n+1)-th iteration of (x + x^2 + x^3) for n>=1.

Original entry on oeis.org

1, 3, 16, 120, 1200, 15078, 228984, 4085028, 83795085, 1943920935, 50333780640, 1439208976920, 45044270036220, 1531759925038616, 56239576979827360, 2217379518189430404, 93441321290076019236, 4191262657895865499821
Offset: 1

Views

Author

Paul D. Hanna, Oct 22 2009

Keywords

Examples

			Let F_n(x) denote the n-th iteration of F(x) = x + x^2 + x^3;
then coefficients in the successive iterations of F(x) begin:
F(x):[1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
F_2: [(1), 2, 4, 6, 8, 8, 6, 3, 1, 0, 0, ...];
F_3: [1, (3), 9, 24, 60, 138, 294, 579, 1053, 1767, 2739, ...];
F_4: [1, 4, (16), 60, 216, 744, 2460, 7818, 23910, 70446, 200160, ...];
F_5: [1, 5, 25, (120), 560, 2540, 11220, 48330, 203230, 835080, ...];
F_6: [1, 6, 36, 210, (1200), 6720, 36930, 199365, 1058175, ...];
F_7: [1, 7, 49, 336, 2268, (15078), 98826, 639093, 4080531, ...];
F_8: [1, 8, 64, 504, 3920, 30128, (228984), 1722084, 12821788, ...];
F_9: [1, 9, 81, 720, 6336, 55224, 477000, (4085028), 34700940, ...];
F_10:[1, 10, 100, 990, 9720, 94680, 915390, 8787735, (83795085), ...]; ...
where the coefficients along the diagonal (shown above in parenthesis)
form the initial terms of this sequence.

Crossrefs

Cf. A166880, A166881, A166882, A166884.

Programs

  • PARI
    {a(n)=local(F=x+x^2+x^3, 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)))}

A166885 Column 1 of triangle A166884.

Original entry on oeis.org

1, 1, 3, 15, 114, 1159, 14838, 229401, 4159662, 86580636, 2034850425, 53303009286, 1539990513588, 48648616439496, 1668228105283302, 61715049142446537, 2450018515737072792, 103892256368706869356, 4686744256645813560957
Offset: 1

Views

Author

Paul D. Hanna, Nov 21 2009

Keywords

Comments

Triangle A166884 transforms diagonals in the triangle A166880 of coefficients in the successive iterations of x+x^2+x^3.

Crossrefs

Cf. A166880, A166884, A166886, A166887.

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+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, 1]}

A166886 Column 2 of triangle A166884.

Original entry on oeis.org

1, 2, 9, 62, 593, 7266, 108720, 1922166, 39212154, 906623004, 23429034168, 669203550906, 20935080981744, 711872134399868, 26142553369667634, 1031146768716808794, 43475757877044427198, 1951261759908828697902
Offset: 1

Views

Author

Paul D. Hanna, Nov 21 2009

Keywords

Comments

Triangle A166884 transforms diagonals in the triangle A166880 of coefficients in the successive iterations of x+x^2+x^3.

Crossrefs

Cf. A166880, A166884, A166885, A166887.

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+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, 2]}

A166887 Column 3 of triangle A166884.

Original entry on oeis.org

1, 3, 18, 157, 1812, 25989, 445255, 8865333, 201058614, 5114874693, 144207579708, 4462151144553, 150316762118466, 5475746846833734, 214463847533104125, 8986421286160678944, 401112805593137715609, 18999650382886046745879
Offset: 1

Views

Author

Paul D. Hanna, Nov 21 2009

Keywords

Comments

Triangle A166884 transforms diagonals in the triangle A166880 of coefficients in the successive iterations of x+x^2+x^3.

Crossrefs

Cf. A166880, A166884, A166885, A166886.

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+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+3, 3]}

A166999 a(n) = a(n-1) + a(n-1)^2 + a(n-1)^3 for n>0 with a(0)=1.

Original entry on oeis.org

1, 3, 39, 60879, 225636660844959, 11487591726386681145142587842614325062822719
Offset: 0

Views

Author

Paul D. Hanna, Nov 21 2009

Keywords

Crossrefs

Cf. A166880.

Programs

  • Mathematica
    NestList[#+#^2+#^3&,1,5] (* Harvey P. Dale, Sep 22 2024 *)
  • PARI
    a(n)=if(n==0,1,a(n-1)+a(n-1)^2+a(n-1)^3)

Formula

Row sums of triangle A166880, which lists the coefficients in the iterations of x+x^2+x^3.

A229112 Row sums of triangle A166884.

Original entry on oeis.org

1, 2, 6, 28, 199, 1945, 24284, 369007, 6606841, 136189033, 3176299055, 82687352399, 2376681846391, 74755785129007, 2554042404290937, 94185081322401217, 3728691027764891142, 157729043279607820306, 7100056927514281702122, 338867203461763515919479
Offset: 0

Views

Author

Paul D. Hanna, Sep 13 2013

Keywords

Comments

Triangle A166884 transforms diagonals in the table of coefficients of successive iterations of x+x^2+x^3 (cf. A166880).

Examples

			Triangle A166884 begins:
1;
1, 1;
3, 2, 1;
15, 9, 3, 1;
114, 62, 18, 4, 1;
1159, 593, 157, 30, 5, 1;
14838, 7266, 1812, 316, 45, 6, 1;
229401, 108720, 25989, 4271, 555, 63, 7, 1; ...
of which the row sums form this sequence.

Crossrefs

Cf. A166884.

Programs

  • PARI
    {a(n, k)=local(F=x, M, N, P, m=max(n, k), A166884); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+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]); A166884=P~*(N~)^-1;sum(k=0,n,A166884[n+1, k+1])}
for(n=0, 25, print1(a(n), ", "))
Showing 1-10 of 10 results.

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