A127648 -id:A127648 - cp's OEIS frontend

A143311 Triangle read by rows, A127648 * A126988; 1<=k<=n.

1, 4, 2, 9, 0, 3, 16, 8, 0, 4, 25, 0, 0, 0, 5, 36, 18, 12, 0, 0, 6, 49, 0, 0, 0, 0, 0, 7, 64, 32, 0, 16, 0, 0, 0, 8, 81, 0, 27, 0, 0, 0, 0, 0, 9, 100, 50, 0, 0, 20, 0, 0, 0, 0, 10, 121, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 144, 72, 48, 36, 0, 24, 0, 0, 0, 0, 0, 12, 169, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13

Offset: 1

Gary W. Adamson, Aug 06 2008

Row sums = A064987, n*sigma(n): (1, 6, 12, 28, 30, 72, 56,...).

			First few rows of the triangle =
1;
4, 2;
9, 0, 3;
16, 8, 0, 4;
25, 0, 0, 0, 5;
36, 18, 12, 0, 0, 6;
49, 0, 0, 0, 0, 0, 7;
64, 32, 0, 16, 0, 0, 0, 8;
...

Cf. A127648, A126988, A064987.

Triangle read by rows, A127648 * A126988; 1<=k<=n

A157497 Triangle read by rows, A156348 * A127648.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 4, 0, 4, 1, 0, 0, 0, 5, 1, 6, 9, 0, 0, 6, 1, 0, 0, 0, 0, 0, 7, 1, 8, 0, 16, 0, 0, 0, 8, 1, 0, 18, 0, 0, 0, 0, 0, 9, 1, 10, 0, 0, 25, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 12, 30, 40, 0, 36, 0, 0, 0, 0, 0, 12

Offset: 1

Gary W. Adamson & Mats Granvik, Mar 01 2009

Row sums = A157020: (1, 3, 4, 9, 6, 22, 8,...)

			First few rows of the triangle =
1;
1, 2;
1, 0, 3;
1, 4, 0, 4;
1, 0, 0, 0, 5;
1, 6, 9, 0, 0, 6;
1, 0, 0, 0, 0, 0, 7;
1, 8, 0, 16, 0, 0, 0, 8;
1, 0, 18, 0, 0, 0, 0, 0, 9;
1, 10, 0, 0, 25, 0, 0, 0, 0, 10;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11;
1, 12, 30, 40, 0, 36, 0, 0, 0, 0, 0, 12;
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
1, 14, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 14;
...
Row 4 = (1, 4, 0, 4) = termwise products of (1, 2, 0, 1) and (1, 2, 3, 4)
where (1, 2, 0, 1) = row 4 of triangle A156348.

Cf. A156348, A126648, A156020

Triangle read by rows, A156348 * A127648. A127648 = an infinite lower triangular matrix with (1, 2, 3,...) as the main diagonal and the rest zeros.

A127954 Triangle, A097805 * A127648.

Original entry on oeis.org

1, 0, 2, 0, 2, 3, 0, 2, 6, 4, 0, 2, 9, 12, 5, 0, 2, 12, 24, 20, 6, 0, 2, 15, 40, 50, 30, 7, 0, 2, 18, 60, 100, 90, 42, 8, 0, 2, 21, 84, 175, 210, 147, 56, 9, 0, 2, 24, 112, 280, 420, 392, 224, 72, 10

Offset: 1

Gary W. Adamson, Feb 09 2007

Row sums = A045623 starting (1, 2, 5, 12, 28, 64, ...).
Dropping the first column gives A128710. - Peter Bala, Mar 05 2013
T(n,k) is the number of ways to place n unlabeled balls into 2 boxes, make compositions of the integer number of balls in each box so that the total number of parts in both compositions is k. - Geoffrey Critzer, Sep 21 2013

			First few rows of the triangle are:
  1;
  0, 2;
  0, 2, 3;
  0, 2, 6, 4;
  0, 2, 9, 12, 5;
  0, 2, 12, 24, 20, 6;
  0, 2, 15, 40, 50, 30, 7;
  ...
T(4,3)=12. Place 4 unlabeled balls into 2 labeled boxes then make compositions of the integer number of balls in each box so that there are a total of 3 parts.
/**** 3 ways since there are 3 compositions of 4 into 3 parts.
*/***  2 ways 1;1+2  and 1;2+1
**/**  2 ways 2;1+1  and 1+1;2.
***/*  2 ways as above.
****/  3 ways as above.
3+2+2+2+3=12. - _Geoffrey Critzer_, Sep 21 2013

Cf. A001263, A097805, A127648, A128710.

nn=10;a=x/(1-x);CoefficientList[Series[1/(1-y a)^2,{x,0,nn}],{x,y}]//Grid (* Geoffrey Critzer, Sep 21 2013 *)

A097805 * A127648 as infinite lower triangular matrices.
E.g.f.: 1/(1 - y*(x/(1-x)))^2. - Geoffrey Critzer, Sep 21 2013
O.g.f.: (1+A001263(x,y))^2, - Vladimir Kruchinin, Oct 15 2020

A143219 Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.

Original entry on oeis.org

1, 2, 6, 3, 9, 18, 4, 12, 24, 40, 5, 15, 30, 50, 75, 6, 18, 36, 60, 90, 126, 7, 21, 42, 70, 105, 147, 196, 8, 24, 48, 80, 120, 168, 224, 288, 9, 27, 54, 90, 135, 189, 252, 324, 405, 10, 30, 60, 100, 150, 210, 280, 360, 450, 550

Offset: 1

Gary W. Adamson, Jul 30 2008

			First few rows of the triangle =
  1;
  2,  6;
  3,  9, 18;
  4, 12, 24, 40;
  5, 15, 30, 50,  75;
  6, 18, 36, 60,  90, 126;
  7, 21, 42, 70, 105, 147, 196;
  ...

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

Cf. A000012, A127648, A127773.

Cf. A002024, A002411 (right border), A002414, A002417 (row sums), A011379.

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

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

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

Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.
Sum_{k=1..n} T(n, k) = A002417(n).
T(n, n) = A002411(n).
From G. C. Greubel, Jul 12 2022: (Start)
T(n, k) = A002024(n,k) * A127773(n,k).
T(n, k) = n * binomial(k+1, 2).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/4)*(4*n - 3*floor((n+1)/2) + 3)*binomial(2 + floor((n+1)/2), 3).
T(2*n-1, n) = A002414(n), n >= 1.
T(2*n-2, n-1) = A011379(n-1), n >= 2. (End)

A143267 Triangle read by rows, A130207 * A000012 * A127648.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 2, 4, 6, 8, 4, 8, 12, 16, 20, 2, 4, 6, 8, 10, 12, 6, 12, 18, 24, 30, 36, 42, 4, 8, 12, 16, 20, 24, 28, 32, 6, 12, 18, 24, 30, 36, 42, 48, 54, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100

Offset: 1

Gary W. Adamson, Aug 03 2008

Left border = phi(n), A000010.

Row sums = A143268, phi(n)*T(n): (1, 3, 12, 20, 60, 42,...)

			First few rows of the triangle =
1;
1, 2;
2, 4, 6;
2, 4, 6, 8;
4, 8, 12, 16, 20;
2, 4, 6, 8, 10, 12;
6, 12, 18, 24, 30, 36, 42;
4, 8, 12, 16, 20, 24, 28, 32;
...
Row 5 = (4, 8, 12, 16, 20) since the first terms of phi(5) = 4; so we perform (4*1, 4*2, 4*3, 4*4, 4*5).

Cf. A130207, A127648, A000010.

Triangle read by rows, A130207 * A000012 * A127648; 1<=k<=n. T(n,k) = phi(n)*k.

A143269 Triangle read by rows, A127648 * A000012 * A130207, 1<=k<=n.

Original entry on oeis.org

1, 2, 2, 3, 3, 6, 4, 4, 8, 8, 5, 5, 10, 10, 20, 6, 6, 12, 12, 24, 12, 7, 7, 14, 14, 28, 14, 42, 8, 8, 16, 16, 32, 16, 48, 32, 9, 9, 18, 18, 36, 18, 54, 36, 54, 10, 10, 20, 20, 40, 20, 60, 40, 60, 40, 11, 11, 22, 22, 44, 22, 66, 44, 66, 44, 110

Offset: 1

Gary W. Adamson, Aug 03 2008

Row sums = A143270: (1, 4, 12, 24, 50, 72, 126, 176,...).

			First few rows of the triangle =
1;
2, 2;
3, 3, 6;
4, 4, 8, 8;
5, 5, 10, 10, 20;
6, 6, 12, 12, 24, 12;
7, 7, 14, 1428, 14, 42;
...
Row 5 = (5, 5, 10, 10, 20) = (5*1, 5*1, 5*2, 5*2, 5*4); where phi(k) = (1, 1, 2, 2, 4,...).

Cf. A000010, A143270.

Triangle read by rows, A127648 * A000012 * A130207. T(n,k) = n*phi(k)

A143271 Triangle read by rows: A130209 * A000012 * A127648.

Original entry on oeis.org

1, 2, 4, 2, 4, 6, 3, 6, 9, 12, 2, 4, 6, 8, 10, 4, 8, 12, 16, 20, 24, 2, 4, 6, 8, 10, 12, 14, 4, 8, 12, 16, 20, 24, 28, 32, 3, 6, 9, 12, 15, 18, 21, 24, 27, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72

Offset: 1

Gary W. Adamson, Aug 03 2008

Row sums = A143272: (1, 6, 12, 30, 30, 84, 56, ...).

Left border = A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, ...).

			First few rows of the triangle =
  1;
  2, 4;
  2, 4,  6;
  3, 6,  9, 12;
  2, 4,  6,  8, 10;
  4, 8, 12, 16, 20, 24;
  2, 4,  6,  8, 10, 12, 14;
  ...
T(5,3) = 6 = 2*3 = d(5)*3.

Cf. A000005, A127648, A130209, A143272.

tabl(nn) = for (n=1, nn, for (k=1, n, print1(numdiv(n)*k, ", "))); \\ Michel Marcus, Jun 05 2023

T(n,k) = d(n)*k.

a(62) corrected by Georg Fischer, Jun 05 2023

A143273 Triangle read by rows: A127648 * A000012 * A130209.

Original entry on oeis.org

1, 2, 4, 3, 6, 6, 4, 8, 8, 12, 5, 10, 10, 15, 10, 6, 12, 12, 18, 12, 24, 7, 14, 14, 21, 14, 28, 14, 8, 16, 16, 24, 16, 32, 16, 32, 9, 18, 18, 27, 18, 36, 18, 36, 27, 10, 20, 20, 30, 20, 40, 20, 40, 30, 40, 11, 22, 22, 33, 22, 44, 22, 44, 33, 44, 22

Offset: 1

Gary W. Adamson, Aug 03 2008

Row sums = A143274: (1, 6, 15, 32, 50, 84, ...).

			First few rows of the triangle =
  1;
  2,  4;
  3,  6,  6;
  4,  8,  8, 12;
  5, 10, 10, 15, 10;
  6, 12, 12, 18, 12, 24;
  7, 14, 14, 21, 14, 28, 14;
  8, 16, 16, 24, 16, 32, 16, 32;
  ...
T(6,3) = 12 = 6*d(3) = 6*2, where A000005 = d(k) = (1, 2, 3, 2, 2, 4, 2, 4, 3, ...).

Cf. A000005, A127648, A130209, A143274.

tabl(nn) = for (n=1, nn, for (k=1, n, print1(n*numdiv(k), ", "))); \\ Michel Marcus, Mar 19 2016

T(n,k) = n*d(k).

a(48) = 20 inserted by Georg Fischer, Jun 05 2023

A143442 Triangle read by rows, A127648 * A000012 * A128407, 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Aug 15 2008

Row sums = A143443, n * (A002321, the Mertens function) = (1, 0, -3, -4, -10, -6, ...).

Right border = n*mu(n).

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

Cf. A002321, A008683, A127648, A128407, A143443.

Triangle read by rows, A127648 * A000012 * A128407, 1 <= k <= n; where A127648 = an infinite lower triangular matrix with (1, 2, 3, ...) in the main diagonal, the rest zeros. A128407 = diagonalized mu(n) matrix.

A134676 A127172 * A127648.

Original entry on oeis.org

1, 3, 2, 3, 0, 3, 6, 6, 0, 4, 3, 0, 0, 0, 5, 9, 6, 9, 0, 0, 6, 3, 0, 0, 0, 0, 0, 7, 10, 12, 0, 12, 0, 0, 0, 8, 6, 0, 9, 0, 0, 0, 0, 0, 9, 9, 6, 0, 0, 15, 0, 0, 0, 0, 10

Offset: 1

Gary W. Adamson, Nov 05 2007

Left column = A007425: (1, 3, 3, 6, 3, 9, 3, 10, ...).

Row sums = A007430: (1, 5, 6, 16, 8, 30, 10, 42, ...).

			First few rows of the triangle:
  1;
  3, 2;
  3, 0, 3;
  6, 6, 0, 4;
  3, 0, 0, 0, 5;
  9, 6, 9, 0, 0, 6;
  3, 0, 0, 0, 0, 0, 7;
  ...

Cf. A127172, A051731, A127648, A007430, A007425.

A127172 * A127648 = A051731^3 * A127648 as infinite lower triangular matrices.

cp's OEIS Frontend

A143311 Triangle read by rows, A127648 * A126988; 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A157497 Triangle read by rows, A156348 * A127648.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A127954 Triangle, A097805 * A127648.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A143219 Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A143267 Triangle read by rows, A130207 * A000012 * A127648.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A143269 Triangle read by rows, A127648 * A000012 * A130207, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A143271 Triangle read by rows: A130209 * A000012 * A127648.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A143273 Triangle read by rows: A127648 * A000012 * A130209.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

Extensions

A143442 Triangle read by rows, A127648 * A000012 * A128407, 1 <= k <= n.

Views

Author