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 49 results. Next

A161363 Inverse of the partition triangle A026794.

Original entry on oeis.org

1, -1, 1, -2, 0, 1, -2, -1, 0, 1, -4, -1, 0, 0, 1, -3, -2, -1, 0, 0, 1, -7, -2, -1, 0, 0, 0, 1, -7, -3, -1, -1, 0, 0, 0, 1, -12, -3, -2, -1, 0, 0, 0, 0, 1, -13, -5, -2, -1, -1, 0, 0, 0, 0, 1, -22, -6, -3, -1, -1, 0, 0, 0, 0, 0, 1, -25, -7, -3, -2, -1, -1, 0, 0, 0, 0, 0, 1, -42, -9, -4, -2, -1, -1, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Jun 07 2009

Keywords

Comments

Row sums = A161375. A modified version of this triangle = A161364.

Examples

			First few rows of the triangle =
1;
-1, 1;
-2, 0, 1;
-2, -1, 0, 1;
-4, -1, 0, 0, 1;
-3, -2, -1, 0, 0, 1;
-7, -2, -1, 0, 0, 0, 1;
-7, -3, -1, -1, 0, 0, 0, 1;
-12, -3, -2, -1, 0, 0, 0, 0, 1;
-13, -5, -2, -1, -1, 0, 0, 0, 0, 1;
-22, -6, -3, -1, -1, 0, 0, 0, 0, 0, 1;
...

Crossrefs

Cf. A000041, A026794, A161364, A161375.

Formula

Triangle read by rows, inverse of A026794.

Extensions

a(3) = 1 corrected and more terms from Georg Fischer, Jun 05 2023

A137585 Triangle read by rows: A054525 * A026794.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 4, 1, 0, 0, 1, 5, 1, 0, 0, 0, 1, 10, 2, 1, 0, 0, 0, 1, 12, 3, 1, 0, 0, 0, 0, 1, 20, 4, 1, 1, 0, 0, 0, 0, 1, 25, 5, 2, 1, 0, 0, 0, 0, 0, 1, 41, 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 47, 10, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1, 76, 14, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Jan 27 2008

Keywords

Comments

Row sums = A000837 starting (1, 1, 2, 3, 6, 7, 14, 17, ...).

Examples

			First few rows of the triangle:
   1;
   0, 1;
   1, 0, 1;
   2, 0, 0, 1;
   4, 1, 0, 0, 1;
   5, 1, 0, 0, 0, 1;
  10, 2, 1, 0, 0, 0, 1;
  12, 3, 1, 0, 0, 0, 0, 1;
  ...

Crossrefs

Cf. A054525, A026794, A000837.

Formula

Mobius transform of A026794, the partition triangle.

A137587 Triangle read by rows: A051731 * A026794.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 5, 2, 0, 1, 6, 1, 0, 0, 1, 11, 3, 2, 0, 0, 1, 12, 2, 1, 0, 0, 0, 1, 20, 6, 1, 2, 0, 0, 0, 1, 25, 4, 3, 1, 0, 0, 0, 0, 1, 37, 9, 2, 1, 2, 0, 0, 0, 0, 1, 43, 8, 3, 1, 1, 0, 0, 0, 0, 0, 1, 70, 16, 6, 3, 1, 2, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Jan 27 2008

Keywords

Comments

That is, regard A051731 and A026794 as lower triangular square matrices and multiply them, then take the lower triangle of the product,
Left column = A083710 starting (1, 2, 3, 5, 6, 11, 12, ...).
Row sums = A047968.

Examples

			First few rows of the triangle:
   1;
   2, 1;
   3, 0, 1;
   5, 2, 0, 1;
   6, 1, 0, 0, 1;
  11, 3, 2, 0, 0, 1;
  12, 2, 1, 0, 0, 0, 1;
  20, 6, 1, 2, 0, 0, 0, 1;
  25, 4, 3, 1, 0, 0, 0, 0, 1;
  ...

Crossrefs

Cf. A026794, A051731, A083710, A047968.

Formula

Inverse mobius transform of the partition triangle, A026794.

Extensions

Typo in 9th row corrected by M. F. Hasler, Jun 08 2009

A137629 Triangle read by rows, A026794^2.

Original entry on oeis.org

1, 2, 1, 4, 0, 1, 7, 2, 0, 1, 11, 2, 0, 0, 1, 18, 4, 2, 0, 0, 1, 26, 4, 2, 0, 0, 0, 1, 39, 9, 2, 2, 0, 0, 0, 1, 55, 9, 4, 2, 0, 0, 0, 0, 1, 79, 16, 4, 2, 2, 0, 0, 0, 0, 1, 106, 18, 6, 2, 2, 0, 0, 0, 0, 0, 1, 150, 29, 9, 4, 2, 2, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Jan 31 2008

Keywords

Comments

Row sums = A137630: (1, 3, 5, 10, 14, 25, 33, 53, 71, 104, ...).
Left border = A137631: (1, 2, 4, 7, 11, 18, 26, 39, 55, 79, ...).

Examples

			First few rows of the triangle:
   1;
   2, 1;
   4, 0, 1;
   7, 2, 0, 1;
  11, 2, 0, 0, 1;
  18, 4, 2, 0, 0, 1;
  26, 4, 2, 0, 0, 0, 1;
  39, 9, 2, 2, 0, 0, 0, 1;
  ...

Crossrefs

Cf. A026794, A137630, A137631.

Formula

Square of the partition triangle.

A137633 Triangle read by rows, A000012 * A026794.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 7, 2, 1, 1, 12, 3, 1, 1, 1, 19, 5, 2, 1, 1, 1, 30, 7, 3, 1, 1, 1, 1, 45, 11, 4, 2, 1, 1, 1, 1, 67, 15, 6, 3, 1, 1, 1, 1, 1, 97, 22, 8, 4, 2, 1, 1, 1, 1, 1, 139, 30, 11, 5, 3, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Gary W. Adamson, Jan 31 2008

Keywords

Comments

Left border = A000070: (1, 2, 4, 7, 12, 19, 30, 45, ...).
Second column = A000041, the partition numbers: (1, 1, 2, 3, 5, 7, 11, ...).
Row sums = A016905: (1, 3, 6, 11, 18, 29, 44, ...).

Examples

			First few rows of the triangle:
   1;
   2, 1;
   4, 1, 1;
   7, 2, 1, 1;
  12, 3, 1, 1, 1;
  19, 5, 2, 1, 1, 1;
  30, 7, 3, 1, 1, 1, 1;
  ...

Crossrefs

Cf. A000041, A000070, A016905, A026794.

Formula

A000012 * A026794 as infinite lower triangular matrices.

A137151 Triangle read by rows: A007318 * A026794.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 13, 4, 3, 1, 34, 9, 6, 4, 1, 88, 22, 11, 10, 5, 1, 225, 55, 22, 20, 15, 6, 1, 569, 137, 50, 36, 35, 21, 7, 1, 1425, 338, 122, 65, 70, 56, 28, 8, 1, 3538, 826, 302, 130, 127, 126, 84, 36, 9, 1, 8717, 2002, 740, 296, 221, 252, 210, 120, 45, 10, 1
Offset: 1

Views

Author

Gary W. Adamson, Jan 23 2008

Keywords

Comments

Row sums = A103446: (1, 3, 8, 21, 54, 137, 344, ...).

Examples

			First few rows of the triangle:
    1;
    2,   1;
    5,   2,  1;
   13,   4,  3,  1;
   34,   9,  6,  4,  1;
   88,  22, 11, 10,  5,  1;
  225,  55, 22, 20, 15,  6, 1;
  569, 137, 50, 36, 35, 21, 7, 1;
  ...

Crossrefs

Cf. A026794, A103446.

Formula

Binomial transform of the partition triangle, A026794.

A137630 Row sums of triangle A137629 (square of the partition triangle A026794).

Original entry on oeis.org

1, 3, 5, 10, 14, 25, 33, 53, 71, 104, 135, 197, 249, 344
Offset: 1

Views

Author

Gary W. Adamson, Jan 31 2008

Keywords

Examples

			a(4) = 10 = sum of row 4 terms of triangle A137629: (7 + 2 + 0 + 1).

Crossrefs

Cf. A026794, A137629.

A137639 Triangle read by rows, A026794 * A051731.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 5, 2, 0, 1, 7, 1, 0, 0, 1, 11, 3, 2, 0, 0, 1, 15, 2, 1, 0, 0, 0, 1, 22, 6, 1, 2, 0, 0, 0, 1, 30, 5, 3, 1, 0, 0, 0, 0, 1, 42, 9, 2, 1, 2, 0, 0, 0, 0, 1, 56, 9, 3, 1, 1, 0, 0, 0, 0, 0, 1, 77, 16, 6, 3, 1, 2, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Jan 31 2008

Keywords

Comments

Left border = A000041 starting with offset 1: (1, 2, 3, 5, 7, 11, 15, ...).
Row sums = A137640: (1, 3, 4, 8, 9, 17, 19, ...).

Examples

			First few rows of the triangle:
   1;
   2, 1;
   3, 0, 1;
   5, 2, 0, 1;
   7, 1, 0, 0, 1;
  11, 3, 2, 0, 0, 1;
  15, 2, 1, 0, 0, 0, 1;
  22, 6, 1, 2, 0, 0, 0, 1;
  30, 5, 3, 1, 0, 0, 0, 0, 1;
  ...

Crossrefs

Cf. A000041, A137640, A026794, A051731.

Formula

A026794 * A051731 as infinite lower triangular matrices.

A137586 Triangle read by rows: A026794 * A054525.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 5, 1, 0, 0, 0, 1, 7, 2, 1, 0, 0, 0, 1, 10, 3, 1, 0, 0, 0, 0, 1, 16, 3, 1, 1, 0, 0, 0, 0, 1, 21, 5, 2, 1, 0, 0, 0, 0, 0, 1, 29, 7, 3, 1, 1, 0, 0, 0, 0, 0, 1, 40, 10, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1, 57, 11, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Jan 27 2008

Keywords

Comments

Row sums = the partition numbers, A000041: (1, 1, 2, 3, 5, 7, 11, 15, ...).
A137585 = A054525 * A026794.

Examples

			First few rows of the triangle:
   1;
   0, 1;
   1, 0, 1;
   2, 0, 0, 1;
   3, 1, 0, 0, 1;
   5, 1, 0, 0, 0, 1;
   7, 2, 1, 0, 0, 0, 1;
  10, 3, 1, 0, 0, 0, 0, 1;
  16, 3, 1, 1, 0, 0, 0, 0, 1;
  ...

Crossrefs

Cf. A000041, A137585, A026794, A054525.

Formula

A026794 * A054525, as infinite lower triangular matrices.
A026794 = the partition triangle, A054525 = the Mobius transform.

A137683 Triangle read by rows, A026794 * A007318^(-1).

Original entry on oeis.org

1, 0, 1, 3, -2, 1, 1, 4, -3, 1, 5, -3, 6, -4, 1, 5, 5, -9, 10, -5, 1, 11, -6, 16, -20, 15, -6, 1, 10, 12, -23, 36, -35, 21, -7, 1, 20, -5, 27, -55, 70, -56, 28, -8, 1, 24, 11, -31, 81, -125, 126, -84, 36, -9, 1, 38, -9, 51, -123, 211, -252, 210, -120, 45, -10, 1
Offset: 0

Views

Author

Gary W. Adamson, Feb 05 2008

Keywords

Comments

Row sums = the partition numbers, A000041: (1, 1, 2, 3, 5, 7, 11, 15, 22, ...).

Examples

			First few rows of the triangle:
  1;
  0,  1;
  3, -2,  1;
  1,  4, -3,  1;
  5, -3,  6, -4,  1;
  5,  5, -9, 10, -5, 1;
  ...
T(3,1) = -3 = (0, 0, 1, -3) dot (3, 1, 0, 1) = (0, 0, 0, -3).

Crossrefs

Cf. A000041, A026794.

Formula

As an infinite lower triangular matrix, A026794 * the inverse of Pascal's triangle.
Showing 1-10 of 49 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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