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.

A166880 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+x^2+x^3).

Original entry on oeis.org

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, 6969, 5961, 4587, 3144, 1896, 990, 438, 159, 45, 9, 1, 1, 4, 16, 60, 216, 744, 2460, 7818, 23910, 70446, 200160, 549006, 1455132, 3730846, 9262712
Offset: 0

Views

Author

Paul D. Hanna, Nov 21 2009

Keywords

Examples

			Triangle begins:
1;
1,1,1;
1,2,4,6,8,8,6,3,1;
1,3,9,24,60,138,294,579,1053,1767,2739,3924,5196,6352,7152,7389,6969,5961,4587,3144,1896,990,438,159,45,9,1;
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,...;
1,9,81,720,6336,55224,477000,4085028,34700940,292495896,...;
1,10,100,990,9720,94680,915390,8787735,83795085,793894860,...;
1,11,121,1320,14300,153890,1645710,17494455,184915225,...;
1,12,144,1716,20328,239448,2805396,32700558,379309986,...;
1,13,169,2184,28080,359268,4575324,58009614,732380298,...;
1,14,196,2730,37856,522704,7188090,98465913,1343828395,...;
1,15,225,3360,49980,740670,10937010,160947465,2360704815,...;
1,16,256,4080,64800,1025760,16185840,254624520,3993857400,...;
1,17,289,4896,82688,1392368,23379216,391488648,6538326616,...;
1,18,324,5814,104040,1856808,33053814,586957419,10398271833,...;
...
The initial diagonals in this triangle 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,...]; ...
The diagonals are transformed one into the other by
triangle A166884, which 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; ...

Crossrefs

Cf. diagonals: A166881, A166882, A166883, related triangle: A166884.
Cf. row sums: A166999, variant: A122888.

Programs

  • PARI
    {T(n, k)=local(F=x+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)))}

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(""))

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)))}
Showing 1-4 of 4 results.

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