A130296 -id:A130296 - cp's OEIS frontend

A131818 A130296 + A002260 - A000012. Triangle read by rows: row n consists of n, 2, 3, 4, ..., n.

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

Offset: 1

Gary W. Adamson, Jul 18 2007

Row sums = A034856; (1, 4, 8, 13, 19, 26, 34, ...).

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

Cf. A130296, A002260, A034856.

Table[Join[{n},Range[2,n]],{n,15}]//Flatten (* Harvey P. Dale, Feb 24 2021 *)

t(n, k) = if (k==1, n, k); \\ Michel Marcus, Feb 12 2014

from math import isqrt, comb
def A131818(n):
    y = (m:=isqrt(k:=n-1<<1))+(k>m*(m+1))
    return n-comb(y,2) # Chai Wah Wu, Jul 07 2025

A130296 + A002260 - A000012 as infinite lower triangular matrices.

T(n, 1) = n, T(n, k) = k for k > 1. - Michel Marcus, Feb 12 2014

More terms from Michel Marcus, Feb 12 2014

A130301 Triangle read by rows: A130296 * A007318, as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 5, 3, 1, 7, 6, 4, 1, 9, 10, 10, 5, 1, 11, 15, 20, 15, 6, 1, 13, 21, 35, 35, 21, 7, 1, 15, 28, 56, 70, 56, 28, 8, 1, 17, 36, 84, 126, 126, 84, 36, 9, 1, 19, 45, 120, 210, 252, 210, 120, 45, 10, 1, 21, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 1

Gary W. Adamson, May 20 2007

Row sums = A083706: (1, 4, 9, 18, 35, 68, ...).
A130300 = A007318 * A130296.
The lower triangular matrix A130296 is equal to the restriction of the square array A051340 to its lower left triangular part. So this is also equal to (A051340) * A007318, where (A051340) is the lower triangular part of A051340, i.e., A051340[i,j] replaced by zero for j > i: see  Mathar's Maple code. - M. F. Hasler, Aug 15 2015

			First few rows of the triangle:
   1;
   3,  1;
   5,  3,  1;
   7,  6,  4,  1;
   9, 10, 10,  5,  1;
  11, 15, 20, 15,  6,  1;
  13, 21, 35, 35, 21,  7,  1;
  ...

Cf. A083706, A130296, A130300.

A130301 := proc(n, k)
    add( A051340(n, i)*binomial(i, k), i=k..n);
end proc: # R. J. Mathar, Jul 16 2015

A130301[m,n] = A121775[m,n] for n >= m/2. A130301[m,1] = 2m-1, A130301[m,2] = A000217[m-1], A130301[m,m] = 1, A130301[m,m-1] = m for m>2. - M. F. Hasler, Aug 15 2015

Corrected (missing a(15)=1 inserted) by M. F. Hasler, Aug 15 2015

a(26) = 27 corrected and more terms from Georg Fischer, May 29 2023

A130303 A130296 * A000012.

Original entry on oeis.org

1, 3, 1, 5, 2, 1, 7, 3, 2, 1, 9, 4, 3, 2, 1, 11, 5, 4, 3, 2, 1, 13, 6, 5, 4, 3, 2, 1, 15, 7, 6, 5, 4, 3, 2, 1, 17, 8, 7, 6, 5, 4, 3, 2, 1, 19, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, May 20 2007

			1;
3, 1;
5, 2, 1;
7, 3, 2, 1;
9, 4, 3, 2, 1;
11, 5, 4, 3, 2, 1;
13, 6, 5, 4, 3, 2, 1;
15, 7, 6, 5, 4, 3, 2, 1;
17, 8, 7, 6, 5, 4, 3, 2, 1;
19, 9, 8, 7, 6, 5, 4, 3, 2, 1;

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp 159-162

Cf. A130296, A000012, A034856 (row sums), A130302 (commuted matrix product)

Clear[e, n, k];
e[n_, 0] := 2*n - 1;
e[n_, k_] := 0 /; k >= n;
e[n_, k_] := (e[n - 1, k]*e[n, k - 1] + 1)/e[n - 1, k - 1];
Table[Table[e[n, k], {k, 0, n - 1}], {n, 1, 10}];
Flatten[%]

A130296 * A000012 as infinite lower triangular matrices. (1,3,5,...) as the left border; (1,2,3,...) in all other columns.

e(n,k)= (e(n - 1, k)*e(n, k - 1) + 1)/e(n - 1, k - 1)

Additional comments from Roger L. Bagula and Gary W. Adamson, Mar 28 2009

A131033 A130296 * A097806.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Jun 10 2007

Row sums give A016777.

A131032 = A097806 * A130296. [corrected by Georg Fischer, Oct 10 2021]

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

Cf. A016777, A097806, A130296.

A130296 * A097806 as infinite lower triangular matrices; where A130296 = (1; 2,1; 3,1,1;...) and A097806 = the pairwise operator.

Definition corrected, a(36)=2 inserted and more terms from Georg Fischer, Oct 10 2021

More terms from Michel Marcus, Oct 11 2021

A130297 A130296^2.

Original entry on oeis.org

1, 4, 1, 8, 2, 1, 13, 3, 2, 1, 19, 4, 3, 2, 1, 26, 5, 4, 3, 2, 1, 34, 6, 5, 4, 3, 2, 1, 43, 7, 6, 5, 4, 3, 2, 1, 53, 8, 7, 6, 5, 4, 3, 2, 1, 64, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, May 20 2007

Left border = A034856: (1, 4, 8, 13, 19, 26, 34, ...).

Row sums = A028387: (1, 5, 11, 19, 29, 41, 55, ...).

			First few rows of the triangle:
   1;
   4, 1;
   8, 2, 1;
  13, 3, 2, 1;
  19, 4, 3, 2, 1;
  26, 5, 4, 3, 2, 1;
  ...

Cf. A130296, A034856, A028387.

from math import comb, isqrt
def A130297(n): return (a:=comb(r:=(m:=isqrt(k:=n<<1))+(k>m*(m+1))+1,2))+1-n+(a-1 if ((k2:=n-1<<1)==(m2:=isqrt(k2))*(m2+1)) else 0) # Chai Wah Wu, Nov 09 2024

Square of A130296 as an infinite lower triangular matrix.

A130298 A051340 * A130296.

Original entry on oeis.org

1, 5, 2, 12, 4, 3, 22, 6, 5, 4, 35, 8, 7, 6, 5, 51, 10, 9, 8, 7, 6, 70, 12, 11, 10, 9, 8, 7, 92, 14, 13, 12, 11, 10, 9, 8, 117, 16, 15, 14, 13, 12, 11, 10, 9, 145, 18, 17, 16, 15, 14, 13, 12, 11, 10

Offset: 1

Gary W. Adamson, May 20 2007

Row sums = A003215: (1, 7, 19, 37, 61, 91, ...).
Left border = A000326: (1, 5, 12, 22, 35, ...).
A130299 = A130296 * A051340.

			First few rows of the triangle:
   1;
   5,  2;
  12,  4,  3;
  22,  6,  5,  4;
  35,  8,  7,  6, 5;
  51, 10,  9,  8, 7, 6;
  70, 12, 11, 10, 9, 8, 7;
  ...

Cf. A051340, A130296, A003215, A000326.

A051340 * A130296 as infinite lower triangular matrices.

A130300 A007318 * A130296.

Original entry on oeis.org

1, 3, 1, 8, 3, 1, 20, 7, 4, 1, 48, 15, 11, 5, 1, 112, 31, 26, 16, 6, 1, 256, 63, 57, 42, 22, 7, 1, 576, 127, 120, 99, 64, 29, 8, 1, 1280, 255, 247, 219, 163, 93, 37, 9, 1, 2816, 511, 502, 466, 382, 256, 130, 46, 10, 1

Offset: 1

Gary W. Adamson, May 20 2007

Row sums = A001787: (1, 4, 12, 32, 80, 192, ...).
Left border = A001792: (1, 3, 8, 20, 48, 112, ...).
A130301 = A130296 * A007318.

			First few rows of the triangle:
    1;
    3,  1;
    8,  3,  1;
   20,  7,  4,  1;
   48, 15, 11,  5,  1;
  112, 31, 26, 16,  6,  1;
  256, 63, 57, 42, 22,  7,  1;
  ...

Cf. A130296, A001787, A001792, A130301.

Binomial transform of A130296.

A130302 A000012 * A130296.

Original entry on oeis.org

1, 3, 1, 6, 2, 1, 10, 3, 2, 1, 15, 4, 3, 2, 1, 21, 5, 4, 3, 2, 1, 28, 6, 5, 4, 3, 2, 1, 36, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, May 20 2007

Row sums = n^2. A130303 = A130296 * A000012.

			First few rows of the triangle are:
1;
3, 1;
6, 2, 1;
10, 3, 2, 1;
15, 4, 3, 2, 1;
21, 5, 4, 3, 2, 1;
...

Cf. A130296, A130303.

A000012 * A130296 as infinite lower triangular matrices. Triangular series as the left border; (1,2,3...) in all other columns.

A131032 A097806 * A130296.

Original entry on oeis.org

1, 3, 1, 5, 2, 1, 7, 2, 2, 1, 9, 2, 2, 2, 1, 11, 2, 2, 2, 2, 1, 13, 2, 2, 2, 2, 2, 1, 15, 2, 2, 2, 2, 2, 2, 1, 17, 2, 2, 2, 2, 2, 2, 2, 1, 19, 2, 2, 2, 2, 2, 2, 2, 2, 1, 21, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 23, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1

Offset: 1

Gary W. Adamson, Jun 10 2007

Row sums give A008574.

A131033 = A130296 * A097806.

			First few rows of the triangle are:
1;
3, 1;
5, 2, 1;
7, 2, 2, 1;
9, 2, 2, 2, 1;
11, 2, 2, 2, 2, 1;
13, 2, 2, 2, 2, 2, 1;
...

Cf. A008574, A097806, A130296, A131033.

A097806 * A130296 as infinite lower triangular matrices. A097806 = the pairwise operator, A130296 = (1; 2,1; 3,1,1; ...).

Definition corrected and more terms from Georg Fischer, Oct 10 2021

A193094 Augmentation of the triangular array P=A130296 whose n-th row is (n+1,1,1,1,1...,1) for 0<=k<=n. See Comments.

Original entry on oeis.org

1, 2, 1, 6, 4, 3, 24, 18, 16, 13, 120, 96, 90, 84, 71, 720, 600, 576, 558, 532, 461, 5040, 4320, 4200, 4128, 4050, 3908, 3447, 40320, 35280, 34560, 34200, 33888, 33462, 32540, 29093, 362880, 322560, 317520, 315360, 313800, 312096, 309330, 302436

Offset: 0

Clark Kimberling, Jul 30 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding W=A193093:
col 1:  A000142, n!
col 2:  A001593, n*n!
col 3:  A130744, n*(n+2)*n!
diag (1,1,3,13,71,...):  A003319, indecomposable permutations.
It appears that T(n,k) is the number of indecomposable permutations p of [n+2] for which p(k+2) = 1. For example, T(2,1) = 4 counts 2413, 3412, 4213, 4312. - David Callan, Aug 27 2014

			First 5 rows:
1
2.....1
6.....4....3
24....18...16...13
120...96...90...84...71

Cf. A193091, A130296, A193093.

p[n_, k_] := If[k == 0, n + 1, 1]
Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A130296 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193094 *)
Flatten[Table[v[n], {n, 0, 9}]]

cp's OEIS Frontend

A131818 A130296 + A002260 - A000012. Triangle read by rows: row n consists of n, 2, 3, 4, ..., n.

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A130301 Triangle read by rows: A130296 * A007318, as infinite lower triangular matrices.

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Maple

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A130303 A130296 * A000012.

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A131033 A130296 * A097806.

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A130297 A130296^2.

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A130298 A051340 * A130296.

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A130300 A007318 * A130296.

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A130302 A000012 * A130296.

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A131032 A097806 * A130296.