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 1-7 of 7 results.

A205548 Symmetric matrix by antidiagonals: C(max(i+1,j+1),min(i+1,j+1)), i>=1, j>=1.

Original entry on oeis.org

1, 3, 3, 6, 1, 6, 10, 4, 4, 10, 15, 10, 1, 10, 15, 21, 20, 5, 5, 20, 21, 28, 35, 15, 1, 15, 35, 28, 36, 56, 35, 6, 6, 35, 56, 36, 45, 84, 70, 21, 1, 21, 70, 84, 45, 55, 120, 126, 56, 7, 7, 56, 126, 120, 55, 66, 165, 210, 126, 28, 1, 28, 126, 210, 165, 66, 78, 220
Offset: 1

Views

Author

Clark Kimberling, Jan 28 2012

Keywords

Examples

			Northwest corner:
1....3....6....10...15
3....1....4....10...20
6....4....1....5....15
10...10...5....1....6
15...20...15...6....1

Crossrefs

Cf. A205456.

Programs

  • Mathematica
    g[k_] := k + 1;
f[i_, j_] := Binomial[Max[g[i], g[j]], Min[g[i], g[j]]]
TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[f[i, n + 1 - i], {n, 1, 14}, {i, 1, n}]]

A205457 Symmetric matrix, by antidiagonals: C(max(2i,2j),min(2i,2j)), i>=0, j>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 6, 6, 1, 1, 15, 1, 15, 1, 1, 28, 15, 15, 28, 1, 1, 45, 70, 1, 70, 45, 1, 1, 66, 210, 28, 28, 210, 66, 1, 1, 91, 495, 210, 1, 210, 495, 91, 1, 1, 120, 1001, 924, 45, 45, 924, 1001, 120, 1, 1, 153, 1820, 3003, 495, 1, 495, 3003, 1820, 153, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Jan 28 2012

Keywords

Examples

			Northwest corner:
1....6....15...28...45
6....1....15...70...210
15...15...1....28...210
28...70...28...1....45
45...210..210..45...1

Crossrefs

Cf. A182878, A205456.

Programs

  • Mathematica
    f[i_, j_] := Binomial[Max[2 i - 2, 2 j - 2], Min[2 i - 2, 2 j - 2]]
TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[f[i, n + 1 - i], {n, 1, 14}, {i, 1, n}]]
  • Maxima
    S(x,y):=((x^3-3*x^2)*y^3-x^2*y^2)/((x^2-2*x+1)*y^3+(-x^2-3)*y^2+(2*x+3)*y-1);
taylor((S(x,y)+S(y,x))/(x*y)-x*y/(1-x*y)+1/(1-x)+1/(1-y)-1,x,0,7,y,0,7); /* Vladimir Kruchinin, Oct 29 2020 */

Formula

G.f.: (S(x,y)+S(y,x))/(x*y)-x*y/(1-x*y)+1/(1-x)+1/(1-y)-1, where S(x,y)=((x^3-3*x^2)*y^3-x^2*y^2)/((x^2-2*x+1)*y^3+(-x^2-3)*y^2+(2*x+3)*y-1). - Vladimir Kruchinin, Oct 29 2020

A205545 Symmetric matrix by antidiagonals: C(max(3i,3j),min(3i,3j)), i>=0, j>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 20, 20, 1, 1, 84, 1, 84, 1, 1, 220, 84, 84, 220, 1, 1, 455, 924, 1, 924, 455, 1, 1, 816, 5005, 220, 220, 5005, 816, 1, 1, 1330, 18564, 5005, 1, 5005, 18564, 1330, 1, 1, 2024, 54264, 48620, 455, 455, 48620, 54264, 2024, 1, 1, 2925
Offset: 1

Views

Author

Clark Kimberling, Jan 28 2012

Keywords

Examples

			Northwest corner:
1.....20....84....220
20....1.....84....924
84....84....1.....220
455...5005..5005..455

Crossrefs

Cf. A205456.

Programs

  • Mathematica
    f[i_, j_] := Binomial[Max[3 i - 3, 3 j - 3], Min[3 i - 3, 3 j - 3]]
TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[f[i, n + 1 - i], {n, 1, 14}, {i, 1, n}]]

A205549 Symmetric matrix by antidiagonals: C(max(i+2,j+2),min(i+2,j+2)), i>=1, j>=1.

Original entry on oeis.org

1, 4, 4, 10, 1, 10, 20, 5, 5, 20, 35, 15, 1, 15, 35, 56, 35, 6, 6, 35, 56, 84, 70, 21, 1, 21, 70, 84, 120, 126, 56, 7, 7, 56, 126, 120, 165, 210, 126, 28, 1, 28, 126, 210, 165, 220, 330, 252, 84, 8, 8, 84, 252, 330, 220, 286, 495, 462, 210, 36, 1, 36, 210, 462
Offset: 1

Views

Author

Clark Kimberling, Jan 28 2012

Keywords

Examples

			Northwest corner:
1....4....10...20...35
4....1....5....15...35
10...5....1....6....21
20...15...6....1....7

Crossrefs

Cf. A205456, A205548.

Programs

  • Mathematica
    g[k_] := k + 2;
f[i_, j_] := Binomial[Max[g[i], g[j]], Min[g[i], g[j]]]
TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[f[i, n + 1 - i], {n, 1, 14}, {i, 1, n}]]

A205550 Symmetric matrix by antidiagonals: C(max(g(i),g(j)),min(g(i),g(j))), where g(k)=2k-1.

Original entry on oeis.org

1, 3, 3, 5, 1, 5, 7, 10, 10, 7, 9, 35, 1, 35, 9, 11, 84, 21, 21, 84, 11, 13, 165, 126, 1, 126, 165, 13, 15, 286, 462, 36, 36, 462, 286, 15, 17, 455, 1287, 330, 1, 330, 1287, 455, 17, 19, 680, 3003, 1716, 55, 55, 1716, 3003, 680, 19, 21, 969, 6188, 6435, 715
Offset: 1

Views

Author

Clark Kimberling, Jan 28 2012

Keywords

Examples

			Northwest corner:
1....3....5....7....9....11
3....1....10...35...84...165
5....10...1....21...126..462
7....35...21...1....36...330

Crossrefs

Cf. A205456.

Programs

  • Mathematica
    g[k_] := 2 k - 1;
f[i_, j_] := Binomial[Max[g[i], g[j]], Min[g[i], g[j]]]
TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[f[i, n + 1 - i], {n, 1, 14}, {i, 1, n}]]

A205552 Square array: C(max(2i-2,j-1),min(2i-2,j-1)), i>=1, j>=1; by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 3, 6, 6, 1, 1, 6, 4, 15, 8, 1, 1, 10, 1, 20, 28, 10, 1, 1, 15, 5, 15, 56, 45, 12, 1, 1, 21, 15, 6, 70, 120, 66, 14, 1, 1, 28, 35, 1, 56, 210, 220, 91, 16, 1, 1, 36, 70, 7, 28, 252, 495, 364, 120, 18, 1, 1, 45, 126, 28, 8, 210, 792, 1001
Offset: 1

Views

Author

Clark Kimberling, Jan 28 2012

Keywords

Examples

			Northwest corner:
1....1....1....1....1
1....2....1....3....6
1....4....6....4....1
1....6....15...20...15
1....8....28...56...70

Crossrefs

Cf. A205456.

Programs

  • Mathematica
    f[i_, j_] := Binomial[Max[2 i - 2, j - 1], Min[2 i - 2, j - 1]]
TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[f[i, n + 1 - i], {n, 1, 14}, {i, 1, n}]]

A205553 Square array by antidiagonals: C(max(i-1,2j-2),min(i-1,2j-2)), i>=1, j>=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 6, 6, 3, 1, 1, 8, 15, 4, 6, 1, 1, 10, 28, 20, 1, 10, 1, 1, 12, 45, 56, 15, 5, 15, 1, 1, 14, 66, 120, 70, 6, 15, 21, 1, 1, 16, 91, 220, 210, 56, 1, 35, 28, 1, 1, 18, 120, 364, 495, 252, 28, 7, 70, 36, 1, 1, 20, 153, 560, 1001, 792, 210, 8
Offset: 1

Views

Author

Clark Kimberling, Jan 28 2012

Keywords

Examples

			Northwest corner:
1....1....1....1....1
1....2....4....6....8
1....1....6....15...28
1....3....4....20...56
1....6....1....15...70

Links

Crossrefs

Cf. A205456.

Programs

  • Mathematica
    f[i_, j_] := Binomial[Max[i - 1, 2 j - 2], Min[i - 1, 2 j - 2]]
TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 10}]]
Flatten[Table[f[i, n + 1 - i], {n, 1, 14}, {i, 1, n}]]
Showing 1-7 of 7 results.

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