A000012 -id:A000012 - cp's OEIS frontend

A137679 Triangle read by rows, A000012 * A008284.

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

Gary W. Adamson, Feb 05 2008

Row sums = A026905: (1, 3, 6, 11, 18, 29, ...).

Rows tend to A000070 starting from the right: (1, 2, 4, 7, 12, 19, 30, ...).

			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;
  ...

Cf. A000070, A008284, A026905.

A000012 * A008284 as infinite lower triangular matrices.

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

A137940 Triangle read by rows, antidiagonals of an array formed by A000012 * A001263 (transform).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 7, 1, 1, 2, 5, 13, 11, 1, 1, 2, 5, 14, 31, 16, 1, 1, 2, 5, 14, 41, 66, 22, 1, 1, 2, 5, 14, 42, 116, 127, 29, 1, 1, 2, 5, 14, 42, 131, 302, 225, 37, 1, 1, 2, 5, 14, 42, 132, 407, 715, 373, 46, 1, 1, 2, 5, 14, 42, 132, 428, 1205, 1549, 586, 56, 1

Offset: 1

Gary W. Adamson, Feb 24 2008

Rows of the array tend to the Catalan sequence, A000108 starting (1, 2, 5, 14, 42, ...).

			First few rows of the array:
  1, 1, 1,  1,  1, ...
  1, 2, 4,  7, 11, ...
  1, 2, 5, 13, 31, ...
  1, 2, 5, 14, 41, ...
  1, 2, 5, 14, 42, ...
  ...
First few rows of the triangle:
  1;
  1, 1;
  1, 2, 1;
  1, 2, 4,  1;
  1, 2, 5,  7,  1;
  1, 2, 5, 13, 11,   1;
  1, 2, 5, 14, 31,  16,   1;
  1, 2, 5, 14, 41,  66,  22,   1;
  1, 2, 5, 14, 42, 116, 127,  29,   1;
  1, 2, 5, 14, 42, 131, 302, 225,  37,  1;
  1, 2, 5, 14, 42, 132, 407, 715, 373, 46, 1;
  ...

Antonio Bernini, Matteo Cervetti, Luca Ferrari, Einar Steingrimsson, Enumerative combinatorics of intervals in the Dyck pattern poset, arXiv:1910.00299 [math.CO], 2019. See Table 1 p. 4.

Cf. A001263, A000108, A106396.

Antidiagonals of an array formed by A000012 * A001263(transform), as infinite triangular matrices. A000012 = (1; 1,1; 1,1,1; 1,1,1,1; ...), A001263 = the Narayana triangle.

More terms from Alois P. Heinz, Nov 28 2021

A140705 A000012 * A051731^4.

Original entry on oeis.org

1, 5, 1, 9, 1, 1, 19, 5, 1, 1, 23, 5, 1, 1, 1, 39, 9, 5, 1, 1, 1, 43, 9, 5, 1, 1, 1, 1, 63, 19, 5, 5, 1, 1, 1, 1, 73, 19, 9, 5, 1, 1, 1, 1, 1, 89, 23, 9, 5, 5, 1, 1, 1, 1, 1, 93, 23, 9, 5, 5, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, May 24 2008

Row sums = A061203: (1, 6, 11, 26, 31, 56,...).

Left column = A061202: (1, 5, 9, 19, 23, 39,...).

			First few rows of the triangle are:
1;
5, 1;
9, 1, 1;
19, 5, 1, 1;
23, 5, 1, 1, 1;
39, 9, 5, 1, 1, 1;
43, 9, 5, 1, 1, 1, 1;
63, 19, 5, 5, 1, 1, 1, 1;
...

Cf. A051731, A061203, A061202.

A000012 * A051731^4 as infinite lower triangular matrices, where A051731 = the inverse Mobius transform and A000012 = an infinite lower triangular matrix with all 1's.

a(60) split into 5,1 by Georg Fischer, Aug 27 2023

A141157 Triangle read by rows, A000012 * A140207.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 4, 3, 5, 4, 6, 6, 5, 6, 5, 8, 9, 10, 7, 7, 6, 10, 12, 15, 14, 11, 8, 7, 12, 15, 20, 21, 22, 15, 9, 8, 14, 18, 25, 28, 33, 30, 22, 10, 9, 16, 21, 30, 35, 44, 45, 44, 30, 11, 10, 18, 24, 35, 42, 55, 60, 66, 60, 42, 12, 11, 20, 27, 40, 49, 66, 75, 88, 90, 84, 56

Offset: 0

Gary W. Adamson & Roger L. Bagula, Jun 12 2008

Right border = partition numbers, A000041.

Row sums = A014153: (1, 3, 7, 14, 26, 45, 75,...).

			First few rows of the triangle are:
1;
2, 1;
3, 2, 2;
4, 3, 4, 3;
5, 4, 6, 6, 5;
6, 5, 8, 9, 10, 7;
7, 6, 10, 12, 15, 14, 11;
8, 7, 12, 15, 20, 21, 22, 15;
9, 8, 14, 18, 25, 28, 33, 30, 22;
10, 9, 16, 21, 30, 35, 44, 45, 44, 30;
...

Cf. A000041, A014153, A140207.

Triangle read by rows, A000012 * A140207; equivalent to taking partial sums of triangle A140207 terms, by columns.

a(60) split and more terms from Georg Fischer, May 29 2023

A143218 Triangle read by rows, A127775 * A000012 * A127775; 1<=k<=n.

Original entry on oeis.org

1, 3, 9, 5, 15, 25, 7, 21, 35, 49, 9, 27, 45, 63, 81, 11, 33, 55, 77, 99, 121, 13, 39, 65, 91, 117, 143, 169, 15, 45, 75, 105, 135, 165, 195, 225, 17, 51, 85, 119, 153, 187, 221, 255, 289, 19, 57, 95, 133, 171, 209, 247, 285, 323, 361, 21, 63, 105, 147, 189, 231, 273, 315, 357, 399, 441

Offset: 1

Gary W. Adamson, Jul 30 2008

			First few rows of the triangle =
   1;
   3,  9;
   5, 15, 25;
   7, 21, 35, 49;
   9, 27, 45, 63,  81;
  11, 33, 55, 77,  99, 121;
  13, 39, 65, 91, 117, 143, 169;
  ...
T(5,3) = 45 = 9*5 = (2*5 - 1) * (2*3 - 1).

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A000012, A005408, A014634, A015237 (row sums).

Cf. A016754, A024598, A033567, A127775, A131507.

[(2*n-1)*(2*k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 12 2022

Table[(2*k-1)*(2*n-1), {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 12 2022 *)

flatten([[(2*n-1)*(2*k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 12 2022

Triangle read by rows, A127775 * A000012 * A127775.
T(n, k) = (2*n - 1) * (2*k - 1), 1<=k<=n.
Sum_{k=1..n} T(n, k) = A015237(n) = n^2 * (2*n-1).
From G. C. Greubel, Jul 12 2022: (Start)
T(n, k) = A131507(n,k) * A127775(n,k).
T(n, n) = A016754(n-1) = (2*n-1)^2, n >= 1.
T(2*n-1, n) = A014634(n-1), n >= 1.
T(2*n-2, n-1) = A033567(n-1), n >= 2.
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A024598(n), n >= 1. (End)

A143230 Triangle read by rows, A130207 * A000012 * A130207.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 2, 2, 4, 4, 4, 4, 8, 8, 16, 2, 2, 4, 4, 8, 4, 6, 6, 12, 12, 24, 12, 36, 4, 4, 8, 8, 16, 8, 24, 16, 6, 6, 12, 12, 24, 12, 36, 24, 36, 4, 4, 8, 8, 16, 8, 24, 16, 24, 16, 10, 10, 20, 20, 40, 20, 60, 40, 60, 40, 100, 4, 4, 8, 8, 16, 8, 24, 16, 24, 16, 40, 16

Offset: 1

Gary W. Adamson, Jul 31 2008

T(n,k) is the number of pairs (a,b), where 0 <= a < n, 0 <= b < k, gcd(a,n) != 1, and gcd(b,k) != 1. - Joerg Arndt, Jun 26 2011

			First few rows of the triangle:
  1;
  1,  1;
  2,  2,  4;
  2,  2,  4,  4;
  4,  4,  8,  8, 16;
  2,  2,  4,  4,  8,  4;
  6,  6, 12, 12, 24, 12, 36;
  4,  4,  8,  8, 16,  8, 24, 16;
  6,  6, 12, 12, 24, 12, 36, 24, 36;
  ...
T(7,5) = 24 = phi(7) * phi(5) = 6 * 4.

Nathaniel Johnston, Rows 1..100, flattened

Cf. A000010, A130207, A143231 (row sums).

A143230:= func< n,k | EulerPhi(n)*EulerPhi(k) >;
[A143230(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Sep 10 2024

with(numtheory): T := proc(n,k) return phi(n)*phi(k): end: seq(seq(T(n,k),k=1..n),n=1..12); # Nathaniel Johnston, Jun 26 2011

A143230[n_, k_]:= EulerPhi[n]*EulerPhi[k];
Table[A143230[n, k], {n, 12}, {k, n}] // Flatten (* G. C. Greubel, Sep 10 2024 *)

def A143230(n,k): return euler_phi(n)*euler_phi(k)
flatten([[A143230(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Sep 10 2024

Triangle read by rows, A130207 * A000012 * A130207, where A130207 = A000010 * 0^(n-k), 1 <= k <= n.
T(n,k) = phi(n) * phi(k), where phi(n) & phi(k) = Euler's totient function.
T(n, 0) = A000010(n) (left border).
Sum_{k=1..n} T(n, k) = A143231(n) (row sums).

A143308 Triangle read by rows, A127446 * A000012, 1<=k<=n.

Original entry on oeis.org

1, 4, 2, 6, 3, 3, 12, 8, 4, 4, 10, 5, 5, 5, 5, 24, 18, 12, 6, 6, 6, 14, 7, 7, 7, 7, 7, 7, 32, 24, 16, 16, 8, 8, 8, 8, 27, 18, 18, 9, 9, 9, 9, 9, 9, 40, 30, 20, 20, 20, 10, 10, 10, 10, 10, 22, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 72, 60, 48, 36, 24, 24, 12, 12, 12, 12, 12, 12

Offset: 1

Gary W. Adamson, Aug 06 2008

Given triangle A127446, partial sums of terms starting from the right.
Row sums give A064987.
Left column is A038040.

			First few rows of the triangle =
   1;
   4,  2;
   6,  3,  3;
  12,  8,  4,  4;
  10,  5,  5,  5,  5;
  24, 18, 12,  6,  6,  6;
  14,  7,  7,  7,  7,  7,  7;
Row 4 = (12, 8, 4, 4) since row 4 of triangle A127446 = (4, 4, 0, 4).

Cf. A038040, A064987, A127446.

a(56) corrected by Georg Fischer, Jul 04 2023

A143349 Triangle read by rows: A000012 * A054524 = A000012 * A051731 * A128407.

Original entry on oeis.org

1, 2, -1, 3, -1, -1, 4, -2, -1, 0, 5, -2, -1, 0, -1, 6, -3, -2, 0, -1, 1, 7, -3, -2, 0, -1, 1, -1, 8, -4, -2, 0, -1, 1, -1, 0, 9, -4, -3, 0, -1, 1, -1, 0, 0, 10, -5, -3, 0, -2, 1, -1, 0, 0, 1, 11, -5, -3, 0, -2, 1, -1, 0, 0, 1, -1, 12, -6, -4, 0, -2, 2, -1, 0, 0, 1, -1, 0, 13, -6, -4, 0, -2, 2, -1, 0, 0, 1, -1, 0, -1, 14, -7, -4, 0, -2, 2, -2, 0, 0, 1, -1, 0, -1, 1

Offset: 1

Gary W. Adamson, Aug 10 2008

The triangle acts as a transform converting any sequence S(k) into a triangle with row sums = S(k). By way of example, begin with S(k), the primes: (2, 3, 5, 7, 11, ...). Add (0, 1, 2, 3, 4, ...) to the sequence getting (prime(n)+(n-1)) = (2, 4, 7, 10, 15, 18, 23, 36, 31, ...) = sequence Q(k). Then replace column 1 (1, 2, 3, ...) of triangle A143349 with sequence Q(k). This = triangle A143350 with row sums prime(n):
    2;
    4, -1;
    7, -1, -1;
   10, -2, -1,  0;
   ...
The A000012 multiplier takes partial sums of A054524 column terms. A051731 is the inverse Mobius transform and A128407 = an infinite lower triangular matrix with mu(n) in the main diagonal and the rest zeros.

			First few rows of the triangle:
   1;
   2, -1;
   3, -1, -1;
   4, -2, -1,  0;
   5, -2, -1,  0, -1;
   6, -3, -2,  0, -1,  1;
   7, -3, -2,  0, -1,  1, -1;
   8, -4, -2,  0, -1,  1, -1,  0;
   9, -4, -3,  0, -1,  1, -1,  0,  0;
  10, -5, -3,  0, -2,  1, -1,  0,  0,  1;
  11, -5, -3,  0, -2,  1, -1,  0,  0,  1, -1;
  12, -6, -4,  0, -2,  2, -1,  0,  0,  1, -1,  0;
  13, -6, -4,  0, -2,  2, -1,  0,  0,  1, -1,  0, -1;
  14, -7, -4,  0, -2,  2, -2,  0,  0,  1, -1,  0, -1,  1;
  ...

Cf. A000012, A051731, A054524, A128407, A143350.

a(39) ff. corrected by Georg Fischer, Jun 05 2023

A143535 Triangle read by rows, A122414 * A000012; 1<=k<=n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Aug 23 2008

Row sums = A008472, the sum of distinct primes dividing n: (0, 2, 3, 2, 5, 5, 7, 2, 3, 7,...). Example: a(10) = 7 = 2 + 5.

			First few rows of the triangle =
0;
1, 1;
1, 1, 1;
1, 1, 0, 0;
1, 1, 1, 1, 1;
2, 2, 1, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1;
1, 1, 0, 0, 0, 0, 0, 0;
1, 1, 1, 0, 0, 0, 0, 0, 0;
2, 2, 1, 1, 1, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
...
Example: row 6 = (2, 2, 1, 0, 0, 0) = partial sums starting from the right of row 6, A122414: (0, 1, 1, 0, 0, 0).

Cf. A122414, A008472.

Triangle read by rows, A122414 * A000012; 1<=k<=n. By rows, partial sums of A122414 terms starting from the right.

A144329 Triangle read by rows, A000012 * A144328.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 4, 3, 5, 4, 6, 6, 4, 6, 5, 8, 9, 8, 5, 7, 6, 10, 12, 12, 10, 6, 8, 7, 12, 15, 16, 15, 12, 7, 9, 8, 14, 18, 20, 20, 18, 14, 8, 10, 9, 16, 21, 24, 25, 24, 21, 16, 9, 11, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10

Offset: 1

Gary W. Adamson, Sep 18 2008

Row sums = A004006: (1, 3, 7, 14, 25, 41, 63, 92, 129,...).

			First few rows of the triangle =
1;
2, 1;
3, 2, 2;
4, 3, 4, 3;
5, 4, 6, 6, 4;
6, 5, 8, 9, 8, 5;
7, 6, 10, 12 12, 10, 6;
8, 7, 12, 15, 16, 15, 12, 7;
...

A144328, Cf. A004006

Triangle read by rows A000012 * A144328.

Partial sums of A144328 by columns.

cp's OEIS Frontend

A137679 Triangle read by rows, A000012 * A008284.

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A137940 Triangle read by rows, antidiagonals of an array formed by A000012 * A001263 (transform).

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A140705 A000012 * A051731^4.

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A141157 Triangle read by rows, A000012 * A140207.

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A143218 Triangle read by rows, A127775 * A000012 * A127775; 1<=k<=n.

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A143230 Triangle read by rows, A130207 * A000012 * A130207.

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A143308 Triangle read by rows, A127446 * A000012, 1<=k<=n.

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A143349 Triangle read by rows: A000012 * A054524 = A000012 * A051731 * A128407.

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