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.

Previous Showing 11-19 of 19 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]}

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]}

A187116 A diagonal of triangle A187115.

Original entry on oeis.org

1, 4, 20, 138, 1258, 14476, 202655, 3354848, 64246776, 1398909332, 34157026526, 924757926320, 27503093599024, 891534666547216, 31289986924954924, 1182213869836721036, 47846600590876178320, 2065282617724684345208
Offset: 2

Views

Author

Paul D. Hanna, Mar 08 2011

Keywords

Comments

Definition of triangle: A187115(n,k) = [y^k] R_{n-1}(y+y^2) for k=2..n, n>2, where R_n(y) is the n-th row polynomial with R_2(y)=y^2.

Crossrefs

Cf. A187115, A135082, A187117, A187118, A187119.

Programs

  • PARI
    {a(n)=local(Rn=y^2);for(m=2,n+1,Rn=subst(truncate(Rn),y,y+y^2+O(y^m)));polcoeff(Rn,n,y)}

Formula

Equals column 1 in the matrix square of triangle A135080.

A187117 A diagonal of triangle A187115.

Original entry on oeis.org

1, 6, 39, 318, 3242, 40348, 598083, 10337402, 204706486, 4577520108, 114208773268, 3147671423964, 95023271024164, 3119560618275648, 110684959354534001, 4221624141483353436, 172274971776090819870
Offset: 2

Views

Author

Paul D. Hanna, Mar 08 2011

Keywords

Comments

Definition of triangle: A187115(n,k) = [y^k] R_{n-1}(y+y^2) for k=2..n, n>2, where R_n(y) is the n-th row polynomial with R_2(y)=y^2.

Crossrefs

Cf. A187115, A135082, A187116, A187118, A187119.

Programs

  • PARI
    {a(n)=local(Rn=y^2);for(m=2,n+2,Rn=subst(truncate(Rn),y,y+y^2+O(y^m)));polcoeff(Rn,n,y)}

Formula

Equals column 1 in the matrix cube of triangle A135080.

A187121 A diagonal of triangle A187120.

Original entry on oeis.org

1, 6, 42, 377, 4248, 58269, 947117, 17848872, 383237040, 9243654925, 247586590398, 7293962774574, 234458181733224, 8167132024738422, 306500617604837898, 12329458457556027538, 529269910501209999900
Offset: 3

Views

Author

Paul D. Hanna, Mar 08 2011

Keywords

Comments

Definition of triangle: A187120(n,k) = [y^k] R_{n-1}(y+y^2) for k=3..n, n>3, where R_n(y) is the n-th row polynomial starting with R_3(y)=y^3.

Crossrefs

Cf. A187120, A135083, A187122, A187123, A187124.

Programs

  • PARI
    {a(n)=local(Rn=y^3);for(m=3,n,Rn=subst(truncate(Rn),y,y+y^2+O(y^m)));polcoeff(Rn,n,y)}

Formula

Equals column 2 in the matrix square of triangle A135080.

A187122 A diagonal of triangle A187120.

Original entry on oeis.org

1, 9, 81, 855, 10758, 159633, 2750067, 54178485, 1204443432, 29871630837, 818490738402, 24571782872034, 802459134168208, 28332664539686670, 1075700621922471621, 43710289920461797346, 1893011243289589171122
Offset: 3

Views

Author

Paul D. Hanna, Mar 08 2011

Keywords

Comments

Definition of triangle: A187120(n,k) = [y^k] R_{n-1}(y+y^2) for k=3..n, n>3, where R_n(y) is the n-th row polynomial starting with R_3(y)=y^3.

Crossrefs

Cf. A187120, A135083, A187121, A187123, A187124.

Programs

  • PARI
    {a(n)=local(Rn=y^3);for(m=3,n+1,Rn=subst(truncate(Rn),y,y+y^2+O(y^m)));polcoeff(Rn,n,y)}

Formula

Equals column 2 in the matrix cube of triangle A135080.

A166904 Row sums of triangle A166900.

Original entry on oeis.org

1, 2, 7, 40, 321, 3361, 43667, 679806, 12358885, 257281501, 6039232167, 157879127902, 4550258562799, 143367509714352, 4903128661348411, 180907738215049666, 7163333648262397913, 303006716530386750233
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, A166903, A122888, A135080.

Programs

  • PARI
    {a(n)=local(F=x, M, N, P); M=matrix(n+2, n+2, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(n+2)))); polcoeff(F, c)); N=matrix(n+1, n+1, 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+1, n+1, r, c, M[r+1, c]); M=(P~*N~^-1); sum(k=1,n+1,M[n+1,k])}
Previous Showing 11-19 of 19 results.

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