A002260 -id:A002260 - cp's OEIS frontend

A208058 Triangle by rows relating to the factorials, generated from A002260.

1, 1, 1, 1, 2, 1, 2, 4, 3, 1, 6, 12, 9, 4, 1, 24, 48, 36, 16, 5, 1, 120, 240, 180, 80, 25, 6, 1, 720, 1440, 1080, 480, 150, 36, 7, 1, 5040, 10080, 7560, 3360, 1050, 252, 49, 8, 1, 40320, 80640, 60480, 26880, 8400, 2016, 392, 64, 9, 1

Offset: 0

Gary W. Adamson, Feb 22 2012

Row sums = A054091: (1, 2, 4, 10, 32, 130, 652,...)

Left border = the factorials, A000142 prefaced with a 1.

			First few rows of the triangle =
1;
1, 1;
1, 2, 1;
2, 4, 3, 1;
6, 12, 9, 4, 1;
24, 48, 36, 16, 5, 1;
120, 240, 180, 80, 25, 6, 1;
720, 1440, 1080, 480, 150, 36, 7, 1;
5040, 10080, 7560, 3360, 1050, 252, 49, 8, 1;
...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000142, A002260, A054091.

T:= proc(n) option remember; local M, k;
      M:= Matrix(n+1, (i, j)->
                 `if`(i=j, 1, `if`(i>j, j*(-1)^(i+j), 0)))^(-1);
      seq(M[n+1, k], k=1..n+1)
    end:
seq(T(n), n=0..14);  # Alois P. Heinz, Feb 24 2012

T[n_] := T[n] = Module[{M}, M = Table[If[i == j, 1, If[i>j, j*(-1)^(i+j), 0]], {i, 1, n+1}, {j, 1, n+1}] // Inverse; M[[n+1]]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

Inverse of:
1;
-1, 1;
1, -2, 1;
-1, 2, -3, 1;
1, -2, 3, -4, 1;
..., where triangle A002260 = (1; 1,2; 1,2,3;...)

A127718 A007318 * A002260 as infinite lower triangular matrices; A002260 = [1; 1,2; 1,2,3; ...].

Original entry on oeis.org

1, 2, 2, 4, 6, 3, 8, 14, 12, 4, 16, 30, 33, 20, 5, 32, 62, 78, 64, 30, 6, 64, 126, 171, 168, 110, 42, 7, 128, 254, 360, 396, 320, 174, 56, 8, 256, 510, 741, 876, 815, 558, 259, 72, 9, 512, 1022, 1506, 1864, 1910, 1536, 910, 368, 90, 10, 1024, 2046, 3039, 3872, 4240

Offset: 1

Gary W. Adamson, Jan 25 2007

Binomial transform of A002260.

Row sums = A084851: (1, 4, 13, 38, 104, 272, ...) A002260 * A007318 = A127717.

			First few rows of the triangle:
   1;
   2,   2;
   4,   6,   3;
   8,  14,  12,   4;
  16,  30,  33,  20,   5;
  32,  62,  78,  64,  30,  6;
  64, 126, 171, 168, 110, 42, 7;
  ...

Cf. A002260, A007318, A084851, A127717.

A007318 := proc(n,k) binomial(n,k) ; end: A002260 := proc(n,k) if k <= n then k; else 0 ; fi ; end: A127718 := proc(n,k) add( A007318(n-1,i-1)*A002260(i,k),i=1..n) ; end: for n from 1 to 15 do for k from 1 to n do printf("%d,",A127718(n,k)) ; od: od: # R. J. Mathar, Oct 02 2007

T(n,k) = Sum_{i=1..n} A007318(n-1,i-1)*A002260(i,k). - R. J. Mathar, Oct 02 2007

More terms from R. J. Mathar, Oct 02 2007

A127735 Triangle read by rows: A127701 * A002260 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 4, 4, 8, 9, 5, 10, 15, 16, 6, 12, 18, 24, 25, 7, 14, 21, 28, 35, 36, 8, 16, 24, 32, 40, 48, 49, 9, 18, 27, 36, 45, 54, 63, 64, 10, 20, 30, 40, 50, 60, 70, 80, 81, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 121, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 144

Offset: 1

Gary W. Adamson, Jan 26 2007

			First few rows of the triangle are:
1;
3, 4;
4, 8, 9;
5, 10, 15, 16;
6, 12, 18, 24, 25;
...

Cf. A002260, A127701, A127736.

Row sums = A127736: (1, 7, 21, 46, 85, 141, ...).

T(n,n) = n^2. T(n,k) = k*(n+1), 1<=kR. J. Mathar, Jul 21 2009

A-number of left factor in the definition corrected by R. J. Mathar, Jul 21 2009

a(19) corrected and more terms from Georg Fischer, Jun 05 2023

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

Gary W. Adamson, Feb 14 2007

Row sums = the triangular numbers: (1, 3, 6, 10, ...).

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

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

Cf. A002260, A128064, A128077, A000326.

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

A130124 Triangle defined by A130123 * A002260, read by rows.

Original entry on oeis.org

1, 2, 4, 4, 8, 12, 8, 16, 24, 32, 16, 32, 48, 64, 80, 32, 64, 96, 128, 160, 192, 64, 128, 192, 256, 320, 384, 448, 128, 256, 384, 512, 640, 768, 896, 1024, 256, 512, 768, 1024, 1280, 1536, 1792, 2048, 2304, 512, 1024, 1536, 2048, 2560, 3072, 3584, 4096, 4608, 5120

Offset: 1

Gary W. Adamson, May 11 2007

Row sums = A001780, (1, 6, 24, 80, 240, ...).

			First few rows of the triangle are:
   1;
   2,  4;
   4,  8, 12;
   8, 16, 24,  32;
  16, 32, 48,  64,  80;
  32, 64, 96, 128, 160, 192; ...

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A001780, A130123, A002260.

Flat(List([1..12], n-> List([1..n], k-> 2^(n-1)*k ))); # G. C. Greubel, Jun 05 2019

[[2^(n-1)*k: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 05 2019

Table[2^(n-1)*k, {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, Jun 05 2019 *)

{T(n,k) = 2^(n-1)*k}; \\ G. C. Greubel, Jun 05 2019

[[2^(n-1)*k for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 05 2019

A130123 * A002260, where A130123 = the 2^n transform and A002260 = [1; 1, 2; 1, 2, 3; ...).

T(n, k) = 2^(n-1)*k. - G. C. Greubel, Jun 05 2019

More terms added by G. C. Greubel, Jun 05 2019

A131416 (A000012 * A002260) + (A002260 * A000012) - A000012.

Original entry on oeis.org

1, 4, 3, 8, 8, 5, 13, 14, 12, 7, 19, 21, 20, 16, 9, 26, 29, 29, 26, 20, 11, 34, 38, 39, 37, 32, 24, 13, 43, 48, 50, 49, 45, 38, 28, 15, 53, 59, 62, 62, 59, 53, 44, 32, 17, 64, 71, 75, 76, 74, 69, 61, 50, 36, 19

Offset: 1

Gary W. Adamson, Jul 08 2007

Left column = A034856: (1, 4, 8, 13, 19, 26, 34,...). Row sums = A127735: (1, 7, 21, 46, 85...).

			First few rows of the triangle are:
1;
4, 3;
8, 8, 5;
13, 14, 12, 7;
19, 21, 20, 16, 9;
26, 29, 29, 26, 20, 11;
...

Cf. A002260, A034856, A127735, A000012.

(A000012 * A002260) + (A002260 * A000012) - A000012; as infinite lower triangular matrices.

A133091 A133080 * A002260.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 2, 4, 6, 4, 1, 2, 3, 4, 5, 2, 4, 6, 8, 10, 6, 1, 2, 3, 4, 5, 6, 7, 2, 4, 6, 8, 10, 12, 14, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2

Offset: 1

Gary W. Adamson, Sep 09 2007

Row sums = A133092: (1, 4, 6, 16, 15, 36, 28, ...).

			First few rows of the triangle:
  1;
  2,  2;
  1,  2,  3;
  2,  4,  6,  4;
  1,  2,  3,  4,  5;
  2,  4,  6,  8, 10,  6;
  1,  2,  3,  4,  5,  6,  7;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A133080, A002260, A133092.

T[n_, n_] := n; T[n_, k_] := (2 - (1 - (-1)^n)/2)*k; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)

for(n=1,10, for(k=1,n, print1(if(k==n, n,(2 - (1 - (-1)^n)/2)*k), ", "))) \\ G. C. Greubel, Oct 21 2017

A133080 * A002260 as infinite lower triangular matrices.

Odd n rows = (1,2,3,...,n). Even n rows = (2,4,6,...,n).

Corrected and extended by Philippe Deléham, Mar 02 2012

A134446 A128174 * A002260.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 4, 3, 4, 3, 4, 6, 4, 5, 3, 6, 6, 8, 5, 6, 4, 6, 9, 8, 10, 6, 7, 4, 8, 9, 12, 10, 12, 7, 8, 5, 8, 12, 12, 15, 12, 14, 8, 9, 5, 10, 12, 16, 15, 18, 14, 16, 9, 10

Offset: 1

Gary W. Adamson, Oct 25 2007

Row sums = A002623: (1, 3, 7, 13, 22, 34, ...).

			First few rows of the triangle:
  1;
  1,  2;
  2,  2,  3;
  2,  4,  3,  4;
  3,  4,  6,  4,  5;
  3,  6,  6,  8,  5,  6;
  4,  6,  9,  8, 10,  6,  7;
  ...

Cf. A002260, A128174, A002623.

A128174 * A002260 as infinite lower triangular matrices.

A134836 Antidiagonals of the array: A007318 * A002260(transposed).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 3, 1, 1, 7, 8, 3, 1, 1, 9, 16, 8, 3, 1, 1, 11, 27, 20, 8, 3, 1, 1, 13, 41, 43, 20, 8, 3, 1, 1, 15, 58, 81, 48, 20, 8, 3, 1, 1, 17, 78, 138, 106, 48, 20, 8, 3, 1, 1, 19, 101, 218, 213, 112, 48, 20, 8, 3, 1

Offset: 1

Gary W. Adamson, Nov 12 2007

Antidiagonals tend to A001792 starting from the right: (1, 3, 8, 20, 48, 112, ...).

			First few rows of the array:
  1,  1,  1,  1,   1,   1, ...;
  1,  3,  3,  3,   3,   3, ...;
  1,  5,  8,  8,   8,   8, ...;
  1,  7, 16, 20,  20,  20, ...;
  1,  9, 27, 43,  48,  48, ...;
  1, 11, 41, 81, 106, 112, ...;
  ...
First few rows of the triangle:
  1;
  1,  1;
  1,  3,  1;
  1,  5,  3,  1;
  1,  7,  8,  3,  1;
  1,  9, 16,  8,  3,  1;
  1, 11, 27, 20,  8,  3,  1;
  1, 13, 41, 43, 20,  8,  3,  1;
  ...

Cf. A002260, A001792, A116445 (array transposed), A001629 (antidiagonal sums).

A002260 := proc(n,k)
    if n <= k then
        n+1;
    else
        0 ;
    end if;
end proc:
A007318 := proc(n,k)
    if k <= n then
        binomial(n,k) ;
    else
        0
    end if;
end proc:
A134836 := proc(n,k)
    add( A007318(n,i)*A002260(i,k),i=0..k) ;
end proc:
seq(seq(A134836(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Aug 17 2022

Antidiagonals of the array: A007318 * A002260(transform), where A002260 = (1; 1,2; 1,2,3; ...).

One term corrected by R. J. Mathar, Aug 17 2022

A134868 A103451 * A002260.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 14 2007

Row sums = A134869: (1, 4, 7, 11, 16, 22, 29, ...).

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

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002260, A071324, A134869.

Table[k + Boole[k == 1 && n != 2], {n, 2, 14}, {k, n - 1}] // Flatten (* Michael De Vlieger, Jul 19 2016 *)

A103451 * A002260 as infinite lower triangular matrices.

Left border of A002260, (1, 1, 1, 1, ...) is replaced by (1, 2, 2, 2, ...).

cp's OEIS Frontend

A208058 Triangle by rows relating to the factorials, generated from A002260.

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A127718 A007318 * A002260 as infinite lower triangular matrices; A002260 = [1; 1,2; 1,2,3; ...].

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A127735 Triangle read by rows: A127701 * A002260 as infinite lower triangular matrices.

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A128078 A002260 * A128064.

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A130124 Triangle defined by A130123 * A002260, read by rows.

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A131416 (A000012 * A002260) + (A002260 * A000012) - A000012.

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A133091 A133080 * A002260.

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A134446 A128174 * A002260.

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