A130478 -id:A130478 - cp's OEIS frontend

A130494 Row sums of triangle A130478.

1, 4, 11, 37, 163, 907, 6067, 47107, 415027, 4084147, 44363827, 526994227, 6793931827, 94451224627, 1408352613427, 22418320792627, 379413423256627, 6802709918872627, 128803497755800627, 2568107879638168627, 53780695151756440627, 1180214324937540760627

Offset: 1

Gary W. Adamson, May 31 2007

nonn

			a(5) = 163 sum of row 5 terms of triangle A130478: (120 + 30 + 8 + 3 + 2); where (30, 8, 3, 2) = the first 4 reversed terms of A001048.
a(5) = 163 = 5! + A130495(4) = 120 + 43.
a(5) = 163 = 5! + (4! + 3!) + (3! + 2!) + (2! + 1!) + (1! + 1).

Cf. A130495, A001048, A130478.

Row sums of triangle A130478. a(1) = 1; a(n), n>1 = n! + A130495(n-1). a(n) = n! + ((n-1)! + (n-2)!) + ((n-2)! + (n-3)!) + ... + (1! + 1). a(n) = n! + sum of first (n-1) terms of A001048 in reverse, where A001048 = (2, 3, 8, 30, 144, ...).

More terms from Alois P. Heinz, Dec 02 2018

A130477 T(n,k) is the number of permutations of [n] with maximum descent k, T(n,k) for n >= 0 and 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 8, 12, 1, 4, 15, 40, 60, 1, 5, 24, 90, 240, 360, 1, 6, 35, 168, 630, 1680, 2520, 1, 7, 48, 280, 1344, 5040, 13440, 20160, 1, 8, 63, 432, 2520, 12096, 45360, 120960, 181440, 1, 9, 80, 630, 4320, 25200, 120960, 453600, 1209600, 1814400, 1, 10, 99, 880, 6930, 47520, 277200, 1330560, 4989600, 13305600, 19958400

Offset: 0

Gary W. Adamson, May 28 2007

Previous name was: Triangle generated from finite differences of A130461.
Decomposition of the permutations.
From Liam Solus, Aug 23 2018: (Start)
For k>0, T(n,k) equals the number of permutations p of [n] such that the largest index i for which p(i)>p(i+1) is k; i.e., T(n,k) is the number of permutations of [n] with maximum descent being k. See Lemma 3.4 of the paper by L. Solus below.
When T(n,k) is taken as the weight of coordinate x_k for k = 0,...,n-1 in an (n-1)-dimensional weighted projective space, the result is the toric variety defined by an n-dimensional simplex whose Ehrhart h^*-polynomial is the n-th Eulerian polynomial. See Theorem 3.5 of the paper by L. Solus below.
(End)

			First few rows of the triangle A130461 = (1; 1, 1; 1, 1, 1; 1, 1, 2, 1; 1, 1, 2, 3, 1; 1, 1, 2, 6, 4, 1;...). Deleting the left border and taking finite differences at the top of each remaining column, we get the first few rows of this triangle:
1;
1, 1;
1, 2,  3;
1, 3,  8,  12;
1, 4, 15,  40,  60;
1, 5, 24,  90, 240,  360;
1, 6, 35, 168, 630, 1680, 2520;
...

L. Solus, Simplices for numeral systems, Transactions of the American Mathematical Society. DOI: https://doi.org/10.1090/tran/7424 (2017).

The triangle with each row reversed is A092582.
Cf. A000142 (row sums), A001710 (main diagonal), A008292.
Cf. A130460, A130461, A130476, A130478.

T := (n,k) -> (n-k+1+0^k)*((n+1)!/(n-k+2)!):
seq(seq(T(n,k),k=0..n),n=0..10); # Peter Luschny, Sep 17 2018

Flatten[Table[Table[(n - k + 1 + 0^k)*(n + 1)!/(n - k + 2)!, {k,0,n}], {n, 0, 10}], 1] (* Olivier Gérard, Aug 04 2012 *)

{T(n, k) = if( n<1 || k>n, 0, k==1, 1, n! * (n+1-k) / (n+2-k)!)}; /* Michael Somos, Jun 25 2017 */

Each term in n-th row divides n!.
Given triangle A130461 and deleting the left border (1,1,1,...) take finite differences by columns and reorient into rows.
T(n,k) = (n-k+1+0^k)*((n+1)!/(n-k+2)!) - Olivier Gérard, Aug 04 2012

New name using a comment by Liam Solus, Peter Luschny, Sep 17 2018

A130460 Infinite lower triangular matrix,(1,0,0,0,...) in the main diagonal and (1,2,3,...) in the subdiagonal.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, May 28 2007

Given M = this sequence as an infinite lower triangular matrix and V = any sequence as a column vector, then M*V is the concatenation of the first term of V with the dot product of (1, 2, 3, ...) and V.

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

Cf. A130461, A130476, A130477, A130478.

A natural number operator as an infinite lower triangular matrix M. (1,0,0,0,...) in the main diagonal, (1,2,3,...) in the subdiagonal and the rest zeros.

a(5) corrected by Gionata Neri, Jun 22 2016

A130493 Triangle read by rows in which row n contains n! repeated n times.

Original entry on oeis.org

1, 2, 2, 6, 6, 6, 24, 24, 24, 24, 120, 120, 120, 120, 120, 720, 720, 720, 720, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 40320, 40320, 40320, 40320, 40320, 40320, 40320, 40320, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880, 362880

Offset: 1

Gary W. Adamson, May 31 2007

Row sums = A001563: (1, 4, 18, 96, 600, 4320, ...). A130477(n,k) * A130478(n,k) = A130493(n,k). Example: take dot products of rows with equal numbers of terms in A130477 and A130478, (1, 3, 8, 12) dot (24, 8, 3, 2) = (24, 24, 24, 24).

			First few rows of the triangle:
   1;
   2,  2;
   6,  6,  6;
  24, 24, 24, 24;
  ...

Cf. A001563, A130477, A130478.

Flatten[Table[Table[n!,{n}],{n,10}]] (* Harvey P. Dale, Dec 24 2014 *)
Table[PadRight[{},n,n!],{n,10}]//Flatten (* Harvey P. Dale, Jul 04 2022 *)

from math import isqrt
from sympy import factorial
def A130493(n): return factorial((m:=isqrt(k:=n<<1))+(k>m*(m+1))) # Chai Wah Wu, Nov 07 2024

Triangle, n! repeated n times per row.

More terms from Sean A. Irvine, Jul 19 2022

A130461 Triangle, antidiagonals of an array generated from A130460.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 6, 4, 1, 1, 1, 2, 6, 12, 5, 1, 1, 1, 2, 6, 24, 20, 6, 1, 1, 1, 2, 6, 24, 60, 30, 7, 1, 1, 1, 2, 6, 24, 120, 120, 42, 8, 1, 1, 1, 2, 6, 24, 120, 360, 210, 56, 9, 1, 1, 1, 2, 6, 24, 120, 720, 840, 336, 72, 10, 1, 1, 1, 2, 6, 24, 120, 720, 2520

Offset: 0

Gary W. Adamson, May 28 2007

Rows tend to the factorials: (1, 1, 2, 6, 24, ...). Row sums = A130476: (1, 2, 3, 5, 8, 15, 28, 61, 132, ...).

			The array =
  1, 1, 1, 1,  1,   1, ...
  1, 1, 2, 3,  4,   5, ...
  1, 1, 2, 6, 12,  20, ...
  1, 1, 2, 6, 24,  60, ...
  1, 1, 2, 6, 24, 120, ...
  1, 1, 2, 6, 24, 120, ...
  ...
First few rows of the triangle:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 2, 1;
  1, 1, 2, 3,  1;
  1, 1, 2, 6,  4,  1;
  1, 1, 2, 6, 12,  5,  1;
  1, 1, 2, 6, 24, 20,  6, 1;
  1, 1, 2, 6, 24, 60, 30, 7, 1;
  ...

Cf. A130460, A130476, A130477, A130478.

Let A130460 = M, an infinite lower triangular matrix and V = [1, 1, 1, ...], the first row of an array. Perform M * V = second row, ...; (n+1)-th row = M * n-th row. The triangle = antidiagonals of the array.

a(23) and a(38) corrected by Gionata Neri, Jun 22 2016

A130476 Row sums of triangle A130461.

Original entry on oeis.org

1, 2, 3, 5, 8, 15, 28, 61, 132, 325, 790, 2133, 5680, 16501

Offset: 1

Gary W. Adamson, May 28 2007

nonn

			a(6) = 15 = (1 + 1 + 2 + 6 + 4 + 1), sum of row 6 terms in triangle A130461.

Cf. A130460, A130461, A130477, A130478.

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A130494 Row sums of triangle A130478.

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A130477 T(n,k) is the number of permutations of [n] with maximum descent k, T(n,k) for n >= 0 and 0 <= k <= n, triangle read by rows.

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A130460 Infinite lower triangular matrix,(1,0,0,0,...) in the main diagonal and (1,2,3,...) in the subdiagonal.

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A130493 Triangle read by rows in which row n contains n! repeated n times.

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A130461 Triangle, antidiagonals of an array generated from A130460.

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A130476 Row sums of triangle A130461.

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