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.

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A135082 Column 1 of triangle A135080.

Original entry on oeis.org

1, 2, 7, 40, 326, 3492, 46558, 744320, 13889080, 296459376, 7125938790, 190502850972, 5607258255032, 180198503713952, 6278311585490032, 235730921392184452, 9489040823468191328, 407662178549724426176
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2007

Keywords

Comments

Triangle A135080 transforms diagonals in the table of coefficients of successive self-compositions of x+x^2 (cf. A122888).

Crossrefs

Cf. A135080 (triangle); other columns: A135081, A135083.

Programs

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

A135083 Column 2 of triangle A135080.

Original entry on oeis.org

1, 3, 15, 112, 1128, 14373, 221952, 4029915, 84135510, 1985740905, 52277994396, 1518768476508, 48261093246396, 1665034362336120, 61979166611850084, 2475861386988907814, 105641851808320785498, 4795101548183135826810
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2007

Keywords

Comments

Triangle A135080 transforms diagonals in the table of coefficients of successive self-compositions of x+x^2 (cf. A122888).

Crossrefs

Cf. A135080 (triangle); other columns: A135081, A135082.

Programs

  • PARI
    {a(n)=local(F=x,M,N,P); M=matrix(n+4,n+4,r,c,F=x;for(i=1,r+c-2,F=subst(F,x,x+x^2+x*O(x^(n+4))));polcoeff(F,c)); N=matrix(n+3,n+3,r,c,M[r,c]);P=matrix(n+3,n+3,r,c,M[r+1,c]);(P~*N~^-1)[n+3,3]}

A187009 G.f. A(x) satisfies: [x^(n+1)] A(F^n(x)) = 0 for n>0 where F^n(x) denotes the n-th iteration of F(x) = x+x^2 with F^0(x)=x.

Original entry on oeis.org

1, -1, 2, -6, 20, -80, 348, -1778, 9892, -64392, 449596, -3609782, 30152616, -284037468, 2694480888, -28592860322, 295151311376, -3440953545088, 37165311149276, -471576198145144, 5062381083026352, -71104461751595892
Offset: 1

Views

Author

Paul D. Hanna, Mar 02 2011

Keywords

Examples

			G.f.: A(x) = x - x^2 + 2*x^3 - 6*x^4 + 20*x^5 - 80*x^6 + 348*x^7 +...
Let F^n(x) denote the n-th iteration of F(x) = x+x^2 with F^0(x)=x,
then the table of coefficients in A(F^n(x)), n>=0, begins:
[1, -1, 2, -6, 20, -80, 348, -1778, 9892, -64392, ...];
[1, 0, 0, -1, 2, -14, 44, -348, 1476, -14148, 73920, ...];
[1, 1, 0, -1, -2, -10, -24, -231, -654, -9276, -32456, ...];
[1, 2, 2, 0, -6, -26, -108, -570, -3216, -22622, -162596, ...];
[1, 3, 6, 8, 0, -54, -324, -1776, -10594, -71702, -540448, ...];
[1, 4, 12, 29, 50, 0, -616, -4846, -32686, -228926, -1749972, ...];
[1, 5, 20, 69, 202, 436, 0, -8629, -84140, -680298, -5508864, ...];
[1, 6, 30, 134, 538, 1880, 4912, 0, -143442, -1672428, -15821492, ...];
[1, 7, 42, 230, 1164, 5404, 22108, 68098, 0, -2762748, -37526484, ...];
[1, 8, 56, 363, 2210, 12646, 67092, 315784, 1122952, 0, -60534272, ..];
[1, 9, 72, 539, 3830, 25930, 166520, 997581, 5322126, 21488640, 0, ..]; ...
in which the main diagonal equals all zeros after the initial '1';
the lower triangular portion of the above table forms triangle A187005.

Crossrefs

Cf. A187005, A135080, A122888, A135081, A187006, A187007, A187008.

Programs

  • PARI
    {ITERATE(F,n,p)=local(G=x);for(i=1,n,G=subst(F,x,G+x*O(x^p)));G}
{a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=-Vec(subst(x*Ser(A),x,ITERATE(x+x^2,i,#A)))[#A]);A[n]}

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)))}

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.

A158830 Triangle, read by rows n>=1, where row n is the n-th differences of column n of array A158825, where the g.f. of row n of A158825 is the n-th iteration of x*Catalan(x).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 5, 1, 0, 0, 14, 10, 0, 0, 0, 42, 70, 8, 0, 0, 0, 132, 424, 160, 4, 0, 0, 0, 429, 2382, 1978, 250, 1, 0, 0, 0, 1430, 12804, 19508, 6276, 302, 0, 0, 0, 0, 4862, 66946, 168608, 106492, 15674, 298, 0, 0, 0, 0, 16796, 343772, 1337684, 1445208, 451948
Offset: 1

Views

Author

Paul D. Hanna, Mar 28 2009

Keywords

Examples

			Triangle begins:
.1;
.1,0;
.2,0,0;
.5,1,0,0;
.14,10,0,0,0;
.42,70,8,0,0,0;
.132,424,160,4,0,0,0;
.429,2382,1978,250,1,0,0,0;
.1430,12804,19508,6276,302,0,0,0,0;
.4862,66946,168608,106492,15674,298,0,0,0,0;
.16796,343772,1337684,1445208,451948,33148,244,0,0,0,0;
.58786,1744314,10003422,16974314,9459090,1614906,61806,162,0,0,0,0;
.208012,8780912,71692452,180308420,161380816,51436848,5090124,103932,84,0,0,0,0;
....
where the g.f. of row n is (1-x)^n*[g.f. of column n of A158825];
g.f. of row n of array A158825 is the n-th iteration of x*C(x):
.1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
.1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
.1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
.1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
.1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
.1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;
.1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;
.1,8,72,684,6720,67620,693048,7209036,75915708,807845676,...;
....
ROW-REVERSAL yields triangle A122890:
.1;
.0,1;
.0,0,2;
.0,0,1,5;
.0,0,0,10,14;
.0,0,0,8,70,42;
.0,0,0,4,160,424,132;
.0,0,0,1,250,1978,2382,429;
.0,0,0,0,302,6276,19508,12804,1430; ...
where g.f. of row n = (1-x)^n*[g.f. of column n of A122888];
g.f. of row n of A122888 is the n-th iteration of x+x^2:
.1;
.1,1;
.1,2,2,1;
.1,3,6,9,10,8,4,1;
.1,4,12,30,64,118,188,258,302,298,244,162,84,32,8,1; ...

Links

Crossrefs

Cf. A158825, A122890 (row-reversal), A122888, columns: A000108, A122892.

Programs

  • Mathematica
    nmax = 11;
f[0][x_] := x; f[n_][x_] := f[n][x] = f[n - 1][x + x^2] // Expand;
T = Table[SeriesCoefficient[f[n][x], {x, 0, k}], {n, 0, nmax}, {k, 1, nmax}];
row[n_] := CoefficientList[(1-x)^n*(T[[All, n]].x^Range[0, nmax])+O[x]^nmax, x] // Reverse;
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Oct 26 2018 *)
  • PARI
    {T(n, k)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P); M=matrix(n+2, n+2, r, c, F=x; for(i=1, r, F=subst(F, x, CAT)); polcoeff(F, c)); Vec(truncate(Ser(vector(n+1,r,M[r,n+1])))*(1-x)^(n+1) +x*O(x^k))[k+1]}

Formula

Row sums equal the factorial numbers.
G.f. of row n = (1-x)^n*[g.f. of column n of A158825] where the g.f. of row n of array A158825 is the n-th iteration of x*C(x) and C(x) is the g.f. of the Catalan sequence A000108.
Row-reversal is triangle A122890 where g.f. of row n of A122890 = (1-x)^n*[g.f. of column n of A122888], and the g.f. of row n of array A122888 is the n-th iteration of x+x^2.

Extensions

Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.

A166901 Column 1 of triangle A166900.

Original entry on oeis.org

1, 4, 21, 156, 1540, 19160, 288813, 5123608, 104657520, 2420186616, 62514944778, 1784255891484, 55767065855228, 1894463658611680, 69504774168222109, 2738952451360200312, 115380142451625516088, 5174227834995200591840
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2009

Keywords

Comments

Triangle A166900 transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

Crossrefs

Cf. A166900, A166902, A166903, A122888, A135080.

Programs

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

A166902 Column 2 of triangle A166900.

Original entry on oeis.org

1, 9, 84, 935, 12480, 196623, 3591560, 74847168, 1755406674, 45804773872, 1317004696656, 41386864224420, 1411592788770580, 51942256939923051, 2051313029747633376, 86548588478842559964, 3885584044838123386104
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2009

Keywords

Comments

Triangle A166900 transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

Crossrefs

Cf. A166900, A166901, A166903, A122888, A135080.

Programs

  • PARI
    {a(n)=local(F=x, M, N, P); M=matrix(n+4, n+4, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(n+4)))); polcoeff(F, c)); N=matrix(n+3, n+3, r, c, F=x;for(i=1, r, F=subst(F, x, x+x^2+x*O(x^(n+4)))); polcoeff(F, c)); P=matrix(n+3, n+3, r, c, M[r+1, c]); (P~*N~^-1)[n+3, 3]}

A166903 Column 3 of triangle A166900.

Original entry on oeis.org

1, 16, 230, 3564, 61845, 1207696, 26415840, 642448632, 17240108314, 506777596248, 16210958231104, 560988459704192, 20891752852722701, 833382707754108896, 35461362393617267808, 1603581518693484768464
Offset: 0

Views

Author

Paul D. Hanna, Nov 27 2009

Keywords

Comments

Triangle A166900 transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

Crossrefs

Cf. A166900, A166901, A166902, A166904, A122888, A135080.

Programs

  • PARI
    {a(n)=local(F=x, M, N, P); M=matrix(n+5, n+5, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(n+5)))); polcoeff(F, c)); N=matrix(n+4, n+4, r, c, F=x;for(i=1, r, F=subst(F, x, x+x^2+x*O(x^(n+5)))); polcoeff(F, c)); P=matrix(n+4, n+4, r, c, M[r+1, c]); (P~*N~^-1)[n+4, 4]}

A122892 Secondary diagonal of triangle A122890.

Original entry on oeis.org

1, 10, 70, 424, 2382, 12804, 66946, 343772, 1744314, 8780912, 43976688, 219546976, 1094132461, 5448766458, 27135655714, 135218356264, 674461583180, 3368432133000, 16847360404260, 84396523333080, 423485559461250
Offset: 4

Views

Author

Paul D. Hanna, Sep 18 2006

Keywords

Comments

The main diagonal of triangle A122890 equals the Catalan numbers (A000108); this is the diagonal immediately below (with leading zeros omitted).

Crossrefs

Cf. A122890, A122888; A122887.
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