cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: José E. Solsona

José E. Solsona's wiki page.

José E. Solsona has authored 4 sequences.

A368396 Prime numbers in Narayana's cows sequence A000930.

Original entry on oeis.org

2, 3, 13, 19, 41, 277, 1873, 8641, 17098272199297
Offset: 1

Views

Author

José E. Solsona, Dec 22 2023

Keywords

Comments

The next terms are too large to show here: a(10) and a(11) have 1036 and 4274 digits, respectively (see A170954). - Jon E. Schoenfield, Dec 29 2023

Crossrefs

Primes in A000930.
Cf. A000040, A170954.

Programs

  • Mathematica
    a[0] = 1; a[1] = a[2] = 1; a[n_] := a[n] = a[n-1] + a[n-3]; Select[Table[a[n],  {n, 0, 400}], PrimeQ]

Formula

a(n) = A000930(A170954(n)).
A000040 INTERSECT A000930.

A362565 The number of linear extensions of n fork-join DAGs of width 4.

Original entry on oeis.org

1, 24, 532224, 237124952064, 765985681152147456, 10915755547826792536473600, 510278911920303453316871670988800, 64243535333922263307871175411271676723200, 18920767554543625469992819764324607588052867481600
Offset: 0

Views

Author

José E. Solsona, Apr 24 2023

Keywords

Comments

The fork-join structure is a modeling structure, commonly seen for example in parallel computing, usually represented as a DAG (or poset). It has an initial "fork" vertex that spawns a number of m independent children vertices (the width) whose output edges are connected to a final "join" vertex. More generally, we can have a number n of these DAGs, each one with m+2 vertices.
When the width is 4 (i.e., m=4), these fork-join DAGs can be depicted as follows (we omit the first column for n=0 because the graph is empty in this case):
n | 1 | 2 | 3
------------------------------------------------------------
| o | o o | o o o
| /| |\ | /| |\ /| |\ | /| |\ /| |\ /| |\
| o o o o | o o o o o o o o | o o o o o o o o o o o o
| \| |/ | \| |/ \| |/ | \| |/ \| |/ \| |/
| o | o o | o o o

Examples

			a(1) = 24 is the number of linear extensions of one fork-join DAG of width 4.

Links

Crossrefs

Row m=4 of A357297.

Programs

  • Mathematica
    a[n_] := (6n)!/30^n
Table[a[n], {n, 0, 8}]

Formula

a(n) = (6n)!/30^n.

A361901 The number of linear extensions of n fork-join DAGs of width 3.

Original entry on oeis.org

1, 6, 9072, 163459296, 15205637551104, 4847253138540933120, 4144575934565485291192320, 8072771848739175726302071357440, 31871690751871005247875440218598277120, 233637150127891005003834299796206474735124480, 2970126289229822074571543766217262582458754059468800
Offset: 0

Views

Author

José E. Solsona, Mar 28 2023

Keywords

Comments

The fork-join structure is a modeling structure, commonly seen for example in parallel computing, usually represented as a DAG (or poset). It has an initial "fork" vertex that spawns a number of m independent children vertices (the width) whose output edges are connected to a final "join" vertex. More generally, we can have a number n of these DAGs, each one with m+2 vertices.
When the width is 3 (i.e. m=3), these fork-join DAGs can be depicted as follows (we omit the first column for n=0 because the graph is empty in this case):
n | 1 | 2 | 3
---------------------------------------------------
| o | o o | o o o
| /|\ | /|\ /|\ | /|\ /|\ /|\
| o o o | o o o o o o | o o o o o o o o o
| \|/ | \|/ \|/ | \|/ \|/ \|/
| o | o o | o o o

Examples

			a(1) = 6 is the number of linear extensions of one fork-join DAG of width 3. Let the DAG be labeled as follows:
     1
   / | \
  2  3  4
   \ | /
     5
Then the six linear extensions are:
  1 2 3 4 5
  1 2 4 3 5
  1 3 2 4 5
  1 3 4 1 5
  1 4 2 3 5
  1 4 3 2 5

Links

Crossrefs

Row m=3 of A357297.

Programs

  • Mathematica
    a[n_] := (5n)!/20^n
Table[a[n], {n, 0, 8}]
  • PARI
    a(n)=(5*n)!/20^n \\ Winston de Greef, Apr 16 2023

Formula

a(n) = (5n)!/20^n.

A357297 T(m,n) is the number of linear extensions of n fork-join DAGs of width m, read by downward antidiagonals.

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 90, 20, 2, 1, 2520, 1680, 280, 6, 1, 113400, 369600, 277200, 9072, 24, 1, 7484400, 168168000, 1009008000, 163459296, 532224, 120, 1, 681080400, 137225088000, 9777287520000, 15205637551104, 237124952064, 49420800, 720, 1, 81729648000, 182509367040000, 207786914375040000, 4847253138540933120, 765985681152147456, 689598074880000, 6671808000, 5040, 1
Offset: 0

Views

Author

José E. Solsona, Feb 22 2023

Keywords

Comments

The fork-join structure is a modeling structure, commonly seen for example in parallel computing, usually represented as a DAG (or poset). It has an initial "fork" vertex that spawns a number of m independent children vertices (the width) whose output edges are connected to a final "join" vertex. More generally, we can have a number n of these DAGs, each one with m+2 vertices.
The family of fork-join DAGs we are considering here can be depicted as follows (we omit the first column for n=0 because the graph is empty in this case):
m\n| 1 | 2 | 3
---------------------------------------------------
0 | o | o o | o o o
| | | | | | | | |
| o | o o | o o o
---------------------------------------------------
| o | o o | o o o
| | | | | | | | |
1 | o | o o | o o o
| | | | | | | | |
| o | o o | o o o
---------------------------------------------------
| o | o o | o o o
| / \ | / \ / \ | / \ / \ / \
2 | o o | o o o o | o o o o o o
| \ / | \ / \ / | \ / \ / \ /
| o | o o | o o o
---------------------------------------------------
| o | o o | o o o
| /|\ | /|\ /|\ | /|\ /|\ /|\
3 | o o o | o o o o o o | o o o o o o o o o
| \|/ | \|/ \|/ | \|/ \|/ \|/
| o | o o | o o o
The array begins like this:
m\n|0 1 2 3 4
-----------------------------------------------------------
0 |1 1 6 90 2520 ... A000680
1 |1 1 20 1680 369600 ... A014606
2 |1 2 280 277200 1009008000 ... A260331
3 |1 6 9072 163459296 15205637551104 ... A361901
4 |1 24 532224 237124952064 765985681152147456 ... A362565
5 |1 120 49420800 689598074880000 97981404549709824000000 ...
with columns: A000012 (n=0) and A000142 (n=1).

Examples

			T(3,1) = 6 is the number of linear extensions of one fork-join DAG of width 3. Let the DAG be labeled as follows:
     1
   / | \
  2  3  4
   \ | /
     5
Then the six linear extensions are:
  1 2 3 4 5
  1 2 4 3 5
  1 3 2 4 5
  1 3 4 1 5
  1 4 2 3 5
  1 4 3 2 5

Links

Crossrefs

Rows m = 0..4 give A000680, A014606, A260331, A361901, A362565.
Columns n = 0..1 give A000012, A000142.

Programs

  • Mathematica
    (* Formula *)
T[m_, n_] := (n*(m+2))!/((m+1)^n*(m+2)^n)
(* 5 X 5 Table *)
Table[T[m, n], {m, 0, 5}, {n, 0, 5}]
(* Eight rows of the triangle *)
Table[Table[T[m, n - m], {m, 0, n}], {n, 0, 8}]
(* As a sequence *)
Flatten[Table[Table[T[m, n - m], {m, 0, n}], {n, 0, 8}]]

Formula

T(m,n) = (n*(m+2))!/((m+1)^n*(m+2)^n).

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