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 22 results. Next

A128077 A128064 * A002260.

Original entry on oeis.org

1, 1, 4, 1, 2, 9, 1, 2, 3, 16, 1, 2, 3, 4, 25, 1, 2, 3, 4, 5, 36, 1, 2, 3, 4, 5, 6, 49, 1, 2, 3, 4, 5, 6, 7, 64
Offset: 1

Views

Author

Gary W. Adamson, Feb 14 2007

Keywords

Comments

Row sums = the pentagonal numbers, A000326: 1, 5, 12, 22, 35, 51, ...

Examples

			First few rows of the triangle:
  1;
  1, 4;
  1, 2, 9;
  1, 2, 3, 16;
  1, 2, 3,  4, 25;
  ...

Crossrefs

Cf. A128064, A002260, A000326, A128078.

Formula

A128064 * A002260 as infinite lower triangular matrices. Triangle read by rows, a(1) = 1; n-th row = first (n-1) terms of (1, 2, 3, ...) followed by n^2.

A153869 Triangle read by rows, A129186 * A128064(unsigned).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 5, 0, 0, 0, 0, 0, 5, 6, 0, 0, 0, 0, 0, 0, 6, 7, 0, 0, 0, 0, 0, 0, 0, 7, 8, 0, 0, 0, 0, 0, 0, 0, 0, 8, 9, 0
Offset: 1

Views

Author

Gary W. Adamson, Jan 03 2009

Keywords

Comments

Lim_{k->inf} A153869^n = A000255: (1, 1, 3, 11, 53, 309, 2119,...).
Row sums = (1, 1, 3, 5, 7, 9,...).
A153869 * (1, 2, 3,...) = A001844 prefaced with a 1: (1, 1, 5, 13, 25, 41,...).

Examples

			First few rows of the triangle =
1;
1, 0;
1, 2, 0;
0, 2, 3, 0;
0, 0, 3, 4, 0;
0, 0, 0, 4, 5, 0;
0, 0, 0, 0, 5, 6, 0;
0, 0, 0, 0, 0, 6, 7, 0;
0, 0, 0, 0, 0, 0, 7, 8, 0;
...

Crossrefs

Cf. A128064, A129186, A000255, A001844

Formula

Triangle read by rows, A129186 * A128064; where A129186 = a shift operator, shifting down triangle A128064(unsigned) one row and inserting a "1" at (1,1).

A128078 A002260 * A128064.

Original entry on oeis.org

1, -1, 4, -1, -2, 9, -1, -2, -3, 16, -1, -2, -3, -4, 25, -1, -2, -3, -4, -5, 36, -1, -2, -3, -4, -5, -6, 49, -1, -2, -3, -4, -5, -6, -7, 64
Offset: 1

Views

Author

Gary W. Adamson, Feb 14 2007

Keywords

Comments

Row sums = the triangular numbers: (1, 3, 6, 10, ...).
Row sums of A128077 = A000326, the pentagonal numbers: (1, 5, 12, 22, 35, ...).

Examples

			First few rows of the triangle:
   1;
  -1,  4;
  -1, -2,  9;
  -1, -2, -3, 16;
  ...

Crossrefs

Cf. A002260, A128064, A128077, A000326.

Formula

A002260 * A128064 as infinite lower triangular matrices. Retain the right border of A128077 and change the signs of all other terms to (-).

A128622 Triangle T(n, k) = A128064(unsigned) * A128174, read by rows.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 3, 4, 3, 4, 5, 4, 5, 4, 5, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 7, 7, 8, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12
Offset: 1

Views

Author

Gary W. Adamson, Mar 14 2007

Keywords

Examples

			First few rows of the triangle are:
  1;
  1, 2;
  3, 2, 3;
  3, 4, 3, 4;
  5, 4, 5, 4, 5;
  5, 6, 5, 6, 5, 6;
  7, 6, 7, 6, 7, 6, 7;
  ...

Links

Crossrefs

Cf. A000027, A016777, A042963, A047461, A109613, A128064, A128174.
Cf. A000326 (diagonal sums), A014848 (row sums), A319556 (alternating row sums).

Programs

  • Magma
    [n - ((n+k) mod 2): k in [1..n], n in [1..16]]; // G. C. Greubel, Mar 14 2024
  • Mathematica
    Table[n - Mod[n+k,2], {n,16}, {k,n}]//Flatten (* G. C. Greubel, Mar 14 2024 *)
  • SageMath
    flatten([[n - ((n+k)%2) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 14 2024

Formula

T(n, k) = abs(A128064(n,k) * A128174(n, k), as infinite lower triangular matrices.
Sum_{k=1..n} T(n, k) = A014848(n) (row sums).
From G. C. Greubel, Mar 14 2024: (Start)
T(n, k) = n - (1 - (-1)^(n+k))/2 = n - (n+k mod 2).
T(n, 1) = A109613(n+1).
T(n, n) = A000027(n).
T(2*n-1, n) = A042963(n).
T(3*n-1, n) = A016777(n+1).
T(4*n-3, n) = A047461(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A319556(n).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A000326(floor((n+1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = A123684(floor((n+1)/2)). (End)

Extensions

More terms added by G. C. Greubel, Mar 14 2024

A132776 A128064 (unsigned) * A007318.

Original entry on oeis.org

1, 3, 2, 5, 8, 3, 7, 18, 15, 4, 9, 32, 42, 24, 5, 11, 50, 90, 80, 35, 6, 13, 72, 165, 200, 135, 48, 7, 15, 98, 273, 420, 385, 210, 63, 8, 17, 128, 420, 784, 910, 672, 308, 80, 9, 19, 162, 612, 1344, 1890, 1764, 1092, 432, 99, 10
Offset: 0

Views

Author

Gary W. Adamson, Aug 29 2007

Keywords

Comments

Row sums = A053220: (1, 5, 16, 44, 112, 272, ...).
A003506 = A007318 * A128064 (unsigned).

Examples

			First few rows of the triangle:
   1;
   3,  2;
   5,  8,  3;
   7, 18, 15,  4;
   9, 32, 42, 24,  5;
  11, 50, 90, 80, 35,  6;
  ...

Crossrefs

Cf. A007318, A128064, A053220, A003506.

Formula

A128064 (unsigned) * A007318 as infinite lower triangular matrices.

A136536 Triangle read by rows: A001263 * A128064 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 5, 7, 3, 14, 19, 19, 4, 42, 51, 71, 41, 5, 132, 146, 216, 216, 76, 6, 429, 449, 617, 827, 547, 127, 7, 1430, 1457, 1793, 2675, 2675, 1205, 197, 8, 4862, 4897, 5497, 8017, 10369, 7429, 2389, 289, 9, 16796, 16840, 17830, 23770, 34858, 34858, 18226, 4366, 406, 10
Offset: 1

Views

Author

Gary W. Adamson, Jan 04 2008

Keywords

Comments

Row sums = A001791: (1, 4, 15, 56, 210, 792, ...).
Left column = A000108 starting (1, 2, 5, 14, 42, 132, 429, ...).

Examples

			First few rows of the triangle:
    1;
    2,   2;
    5,   7,   3;
   14,  19,  19,   4;
   42,  51,  71,  41,   5;
  132, 146, 216, 216,  76,   6;
  429, 449, 617, 827, 547, 127,   7;
  ...

Crossrefs

Cf. A000108, A001263, A001791, A128064.

Extensions

a(46) = 16796 corrected and two more terms from Georg Fischer, May 31 2023

A128065 Binomial transform of A128064.

Original entry on oeis.org

1, 0, 2, -1, 2, 3, -2, 0, 6, 4, -3, -4, 6, 12, 5, -4, -10, 0, 20, 20, 6, -5, -18, -15, 20, 45, 30, 7, -6, -28, -42, 0, 70, 84, 42, 8, -7, -40, -84, -56, 70, 168, 140, 56, 9, -8, -54, -144, -168, 0, 252, 336, 216, 72, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 14 2007

Keywords

Comments

A128064 * A007318 = A103406 row sums = 2^(n-1).

Examples

			First few rows of the triangle are:
   1;
   0,  2;
  -1,  2,  3;
  -2,  0,  6,  4;
  -3, -4,  6, 12,  5;
  ...

Crossrefs

Cf. A128064, A103406.

Programs

  • Magma
    /* As triangle */ [[(k+1)*(Binomial(n,k)-Binomial(n,k+1)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 12 2019
  • Mathematica
    Table[(k + 1) (Binomial[n, k] - Binomial[n, k + 1]), {n, 0,
12}, {k, 0, n}] // Flatten (* Nathaniel MacFadden, Jul 11 2019 *)

Formula

A007318 * A128064 as infinite lower triangular matrices.

Extensions

a(43) corrected by Nathaniel MacFadden, Jul 11 2019

A136534 A001263 * A128064 (unsigned).

Original entry on oeis.org

1, 2, 2, 4, 8, 3, 7, 24, 21, 4, 11, 60, 90, 44, 5, 16, 130, 300, 260, 80, 6, 22, 2252, 840, 1120, 630, 132, 7, 29, 448, 2058, 3920, 3430, 1344, 203, 8, 37, 744, 4536, 11760, 14700, 9072, 2604, 296, 9, 46, 1170, 9180, 31248, 52920, 46872, 21420, 4680, 414, 10
Offset: 1

Views

Author

Gary W. Adamson, Jan 04 2008

Keywords

Comments

Row sums = A001791: (1, 4, 15, 56, 210, ...).
Left column = A000124: (1, 2, 4, 7, 11, 16, 22, ...).

Examples

			First few rows of the triangle:
   1;
   2,   2;
   4,   8,   3;
   7,  24,  21,    4;
  11,  60,  90,   44,   5;
  16, 130, 300,  260,  80,   6;
  22, 252, 840, 1120, 630, 132, 7;
  ...

Crossrefs

Cf. A001263, A128064, A001791, A000124.

Formula

A001263 * A128064 (unsigned), A001263 = the Narayana triangle, A128064 = an infinite lower triangular matrix with (1, 2, 3, ...) in the main and subdiagonals.

A136535 A128064 * A001263.

Original entry on oeis.org

1, 1, 2, 1, 7, 3, 1, 15, 21, 4, 1, 26, 76, 46, 5, 1, 40, 200, 250, 85, 6, 1, 57, 435, 925, 645, 141, 7, 1, 77, 833, 2695, 3185, 1421, 217, 8, 1, 100, 1456, 6664, 11956, 9016, 2800, 316, 9, 1, 126, 2376, 14616, 37044, 42336, 22176, 5076, 441, 10
Offset: 1

Views

Author

Gary W. Adamson, Jan 04 2008

Keywords

Comments

Row sums give A076540.

Examples

			First few rows of the triangle are:
1;
1, 2;
1, 7, 3;
1, 15, 21, 4;
1, 26, 76, 46, 5;
1, 40, 200, 250, 85, 6;
1, 57, 435, 925, 645, 141, 7;
...

Crossrefs

Cf. A001263, A128064, A076540.

Programs

  • PARI
    T4(n,k) = sum(j=k, n, binomial(n,j)*binomial(j,k)*(-1)^(j-k)*(j+1));
T3(n,k) = binomial(n, k)*binomial(n-1, k-1) - binomial(n, k-1)*binomial(n-1, k);
N=10; matrix(N, N, n, k, T4(n-1,k-1))*matrix(N, N,n,k,T3(n,k)) \\ Michel Marcus, Oct 11 2021

Extensions

a(18) corrected by Georg Fischer, Oct 10 2021

A128116 A128064 * A122432 (unsigned).

Original entry on oeis.org

1, 5, 2, 12, 7, 3, 22, 15, 9, 4, 35, 26, 18, 11, 5, 51, 40, 30, 21, 13, 6, 70, 57, 45, 34, 24, 15, 7, 92, 77, 63, 50, 38, 27, 17, 8, 117, 100, 84, 69, 55, 42, 30, 19, 9, 145, 126, 108, 91, 75, 60, 46, 33, 21, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 14 2007

Keywords

Comments

Left border of the triangle = A000326, the pentagonal numbers: (1, 5, 12, 22, 35, ...).
Row sums = A002412: (1, 7, 22, 50, 95, ...).

Examples

			First few rows of the triangle:
   1;
   5,  2;
  12,  7,  3;
  22, 15,  9,  4;
  35, 26, 18, 11,  5;
  51, 40, 30, 21, 13,  6;
  70, 57, 45, 34, 24, 15,  7;
  ...

Crossrefs

Cf. A128064, A122432, A000326, A002412.

Formula

A128064 * A122432 (unsigned), where the unsigned version of A122432 = (1; 3, 1; 6, 3, 1; 10, 6, 3, 1; ...).
Showing 1-10 of 22 results. Next

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

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