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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A050300 Matrix 6th power of partition triangle A008284.

Original entry on oeis.org

1, 6, 1, 21, 6, 1, 71, 27, 6, 1, 196, 92, 27, 6, 1, 532, 288, 98, 27, 6, 1, 1301, 799, 309, 98, 27, 6, 1, 3101, 2100, 891, 315, 98, 27, 6, 1, 6956, 5145, 2373, 912, 315, 98, 27, 6, 1, 15217, 12121, 5980, 2465, 918, 315, 98, 27, 6, 1, 31951, 27247, 14292, 6253
Offset: 1

Views

Author

Christian G. Bower, Aug 15 1999

Keywords

Examples

			1; 6,1; 21,6,1; 71,27,6,1; ...

Crossrefs

Cf. A038497, A038498, A039805-A039807. A050301-A050304. a(n, 1) = A022814(n) (first column).

A055885 Euler transform applied twice to partition triangle A008284.

Original entry on oeis.org

1, 1, 3, 1, 3, 6, 1, 6, 9, 14, 1, 6, 18, 23, 27, 1, 9, 27, 54, 57, 58, 1, 9, 39, 87, 140, 131, 111, 1, 12, 51, 150, 259, 353, 295, 223, 1, 12, 69, 210, 470, 702, 832, 637, 424, 1, 15, 84, 314, 749, 1379, 1803, 1917, 1350, 817, 1, 15, 105, 416, 1176, 2352, 3730, 4403, 4245, 2789, 1527
Offset: 1

Views

Author

Christian G. Bower, Jun 09 2000

Keywords

Examples

			  1;
  1, 3;
  1, 3,  6;
  1, 6,  9, 14;
  1, 6, 18, 23, 27;
  ...

Links

Crossrefs

Row sums give A007713.
Main diagonal gives A001970.
Cf. A055884, A055886.

A055886 Euler transform applied three times to partition triangle A008284.

Original entry on oeis.org

1, 1, 4, 1, 4, 10, 1, 8, 16, 30, 1, 8, 32, 54, 75, 1, 12, 48, 128, 176, 206, 1, 12, 70, 210, 443, 535, 518, 1, 16, 92, 362, 842, 1485, 1585, 1344, 1, 16, 124, 516, 1544, 3075, 4676, 4527, 3357, 1, 20, 152, 770, 2500, 6133, 10622, 14336, 12664, 8429, 1, 20, 190, 1030, 3952, 10718, 22524, 34918, 42426, 34631, 20759
Offset: 1

Views

Author

Christian G. Bower, Jun 09 2000

Keywords

Examples

			  1;
  1, 4;
  1, 4, 10;
  1, 8, 16, 30;
  1, 8, 32, 54, 75;
  ...

Links

Crossrefs

Row sums give A007714.
Main diagonal gives A007713.
Cf. A055884, A055885.

A039806 Matrix 4th power of partition triangle A008284.

Original entry on oeis.org

1, 4, 1, 10, 4, 1, 26, 14, 4, 1, 55, 36, 14, 4, 1, 121, 91, 40, 14, 4, 1, 237, 202, 101, 40, 14, 4, 1, 468, 439, 238, 105, 40, 14, 4, 1, 867, 887, 524, 248, 105, 40, 14, 4, 1, 1597, 1758, 1105, 560, 252, 105, 40, 14, 4, 1, 2821, 3325, 2223, 1190, 570, 252, 105, 40
Offset: 0

Views

Author

Christian G. Bower, Feb 15 1999

Keywords

Comments

Row sums form A022813 (number of terms in n-th derivative of a function composed with itself 5 times). - Paul D. Hanna, Jul 13 2004

Examples

			1; 4,1; 10,4,1; 26,14,4,1; ...

Crossrefs

Cf. A038497, A038498, A039805, A039807. a(n, 1) = A022812(n) (first column).
Cf. A022813.

A050301 Matrix 7th power of partition triangle A008284.

Original entry on oeis.org

1, 7, 1, 28, 7, 1, 105, 35, 7, 1, 322, 133, 35, 7, 1, 952, 455, 140, 35, 7, 1, 2541, 1379, 483, 140, 35, 7, 1, 6539, 3920, 1512, 490, 140, 35, 7, 1, 15833, 10375, 4354, 1540, 490, 140, 35, 7, 1, 37148, 26243, 11803, 4487, 1547, 490, 140, 35, 7, 1, 83594
Offset: 1

Views

Author

Christian G. Bower, Aug 15 1999

Keywords

Examples

			1; 7,1; 28,7,1; 105,35,7,1; ...

Crossrefs

Cf. A038497, A038498, A039805-A039807. A050300-A050304. a(n, 1) = A024207(n) (first column).

A026819 a(n) = least k such that if 1<=h<=n then T(n,k)>=T(n,h), T given by A008284.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Max{T(n, k)} for 1<=k<=n is A002569.

Programs

  • Mathematica
    f[n_] := Block[{k = 1, mk = mx = 0}, While[k < n + 1, a = Length@ IntegerPartitions[n, {k}]; If[a > mx, mx = a; mk = k]; k++ ]; mk]; Array[f, 85] (* Robert G. Wilson v, Jul 20 2010 *)

Extensions

More terms from Robert G. Wilson v, Jul 20 2010

A050303 Matrix 9th power of partition triangle A008284.

Original entry on oeis.org

1, 9, 1, 45, 9, 1, 201, 54, 9, 1, 735, 246, 54, 9, 1, 2517, 981, 255, 54, 9, 1, 7785, 3453, 1026, 255, 54, 9, 1, 22857, 11238, 3699, 1035, 255, 54, 9, 1, 63024, 33930, 12183, 3744, 1035, 255, 54, 9, 1, 166819, 97038, 37464, 12429, 3753, 1035, 255, 54, 9, 1
Offset: 1

Views

Author

Christian G. Bower, Aug 15 1999

Keywords

Examples

			1; 9,1; 45,9,1; 201,54,9,1; ...

Crossrefs

Cf. A038497, A038498, A039805-A039807. A050300-A050304. a(n, 1) = A024209(n) (first column).

A055888 Invert transform of partition triangle A008284.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 5, 8, 8, 1, 6, 16, 20, 16, 1, 8, 25, 46, 48, 32, 1, 9, 37, 84, 124, 112, 64, 1, 11, 50, 142, 256, 320, 256, 128, 1, 12, 67, 216, 480, 732, 800, 576, 256, 1, 14, 84, 319, 812, 1500, 2000, 1952, 1280, 512, 1, 15, 105, 443, 1304, 2772, 4432, 5280
Offset: 1

Views

Author

Christian G. Bower, Jun 09 2000

Keywords

Examples

			1; 1,2; 1,3,4; 1,5,8,8; 1,6,16,20,16; ...

Links

Crossrefs

Row sums give A055887.

A055893 Inverse Moebius transform of partition triangle A008284.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 3, 1, 3, 1, 2, 2, 1, 2, 1, 4, 4, 3, 1, 4, 1, 3, 4, 3, 2, 1, 2, 1, 5, 5, 8, 3, 3, 1, 4, 1, 4, 8, 6, 5, 4, 2, 1, 3, 1, 6, 8, 11, 8, 7, 3, 3, 1, 4, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 2, 1, 7, 13, 19, 13, 17, 7, 8, 4, 3, 1, 6, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 2, 1, 8, 16, 26
Offset: 1

Views

Author

Christian G. Bower, Jun 09 2000

Keywords

Examples

			1; 1,2; 1,1,2; 1,3,1,3; 1,2,2,1,2; ...

Links

Crossrefs

Row sums give A047968. Cf. A055892.

A137679 Triangle read by rows, A000012 * A008284.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 6, 4, 2, 1, 6, 9, 7, 4, 2, 1, 7, 12, 11, 7, 4, 2, 1, 8, 16, 16, 12, 7, 4, 2, 1, 9, 20, 23, 18, 12, 7, 4, 2, 1, 10, 25, 31, 27, 19, 12, 7, 4, 2, 1, 11, 30, 41, 38, 29, 19, 12, 7, 4, 2, 1, 12, 36, 53, 53, 42, 30, 19, 12, 7, 4, 2, 1
Offset: 1

Views

Author

Gary W. Adamson, Feb 05 2008

Keywords

Comments

Row sums = A026905: (1, 3, 6, 11, 18, 29, ...).
Rows tend to A000070 starting from the right: (1, 2, 4, 7, 12, 19, 30, ...).

Examples

			First few rows of the triangle:
  1;
  2,  1;
  3,  2,  1;
  4,  4,  2,  1;
  5,  6,  4,  2, 1;
  6,  9,  7,  4, 2, 1;
  7, 12, 11,  7, 4, 2, 1;
  8, 16, 16, 12, 7, 4, 2, 1;
  ...

Crossrefs

Cf. A000070, A008284, A026905.

Formula

A000012 * A008284 as infinite lower triangular matrices.

Extensions

a(38) = 20 corrected by Georg Fischer, May 29 2023
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