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.

Showing 1-7 of 7 results.

A344112 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not functions.

Original entry on oeis.org

1, 3, 12, 7, 56, 485, 15, 240, 4015, 65280, 31, 992, 32525, 1047552, 33551307, 63, 4032, 261415, 16773120, 1073726199, 68719430080, 127, 16256, 2094965, 268419072, 34359660243, 4398046231168, 562949952597769, 255, 65280, 16770655, 4294901760, 1099511237151
Offset: 1

Views

Author

Mohammad K. Azarian, May 10 2021

Keywords

Examples

			T(2,2) = (number of relations) - (number of functions) = 2^4 - 4 = 12.
Triangle T(n,k) begins:
   1;
   3,  12;
   7,  56,   485;
  15, 240,  4015,   65280;
  31, 992, 32525, 1047552, 33551307;

Links

Crossrefs

Cf. A101030, A199656, A036679, A344110.

Programs

  • Mathematica
    Column[Table[2^(n*k) - k^n, {n, 10}, {k, n}], Center]

Formula

T(n,k) = 2^(n*k) - k^n, n,k >= 1.

A344113 a(n) = 2^(n^2) - n^n.

Original entry on oeis.org

1, 12, 485, 65280, 33551307, 68719430080, 562949952597769, 18446744073692774400, 2417851639229257961991863, 1267650600228229401486703205376, 2658455991569831745807613835249018541, 22300745198530623141535718272639445405532160
Offset: 1

Views

Author

Mohammad K. Azarian, May 14 2021

Keywords

Comments

a(n) is the number of relations on a set with n elements that are not functions.

Examples

			a(1) = 2^(1^2) - 1^1 = 1.
a(2) = 2^(2^2) - 2^2 = 12.
a(3) = 2^(3^2) - 3^3 = 485.

Links

Crossrefs

Cf. A000312, A002416, A036679, A344110, A344112, A344114.

Programs

  • Mathematica
    Table[2^(n^2) - n^n, {n, 12}] // Flatten

A344114 a(n) = 2^(n^2) - n!.

Original entry on oeis.org

1, 14, 506, 65512, 33554312, 68719476016, 562949953416272, 18446744073709511296, 2417851639229258349049472, 1267650600228229401496699576576, 2658455991569831745807614120520772352, 22300745198530623141535718272648361026978816, 748288838313422294120286634350736906063831234982912
Offset: 1

Views

Author

Mohammad K. Azarian, Jun 04 2021

Keywords

Comments

a(n) is the number of relations on a set with n elements that are not one-to-one functions.

Examples

			a(1) = 2^(1^2) - 1! =   1;
a(2) = 2^(2^2) - 2! =  14;
a(3) = 2^(3^2) - 3! = 506.

Links

Crossrefs

Cf. A000312, A002416, A036679, A101030, A199656, A344110, A344112, A344113.

Programs

  • Mathematica
    Table[2^(n^2) - n!, {n, 16}] // Flatten

A344115 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not one-to-one functions.

Original entry on oeis.org

1, 2, 14, 5, 58, 506, 12, 244, 4072, 65512, 27, 1004, 32708, 1048456, 33554312, 58, 4066, 262024, 16776856, 1073741104, 68719476016, 121, 16342, 2096942, 268434616, 34359735848, 4398046506064, 562949953416272, 248, 65480, 16776880, 4294965616, 1099511621056
Offset: 1

Views

Author

Mohammad K. Azarian, Jun 06 2021

Keywords

Comments

If n=k, then T(n,k) = 2^(n^2) - n!, which is A344114, and if kA344110.

Examples

			For T(2,2): the number of relations is 2^4 and the number of one-to-one functions is 2, so 2^4 - 2 = 14 and thus T(2,2) = 14.
Triangle T(n,k) begins:
   1;
   2,   14;
   5,   58,   506;
  12,  244,  4072,   65512;
  27, 1004, 32708, 1048456, 33554312;

Links

Crossrefs

Cf. A000312, A002416, A036679, A068424, A101030, A199656, A344110, A344112, A344113, A344114.

Programs

  • Mathematica
    Table[2^(n*k) - k!/(k - n)!, {k, 10}, {n, k}] // Flatten

Formula

T(n,k) = 2^(n*k) - k!/(k-n)!, k >= n.

A344116 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not onto functions.

Original entry on oeis.org

1, 3, 14, 7, 58, 506, 15, 242, 4060, 65512, 31, 994, 32618, 1048336, 33554312, 63, 4034, 261604, 16775656, 1073740024, 68719476016, 127, 16258, 2095346, 268427056, 34359721568, 4398046495984, 562949953416272, 255, 65282, 16771420, 4294926472, 1099511501776, 281474976519136, 72057594037786816, 18446744073709511296
Offset: 1

Views

Author

Mohammad K. Azarian, Jun 07 2021

Keywords

Examples

			For T(2,2), the number of relations is 2^4 and the number of onto functions is 2, so 2^4 - 2 = 14.
Triangle T(n,k) begins:
   1
   3     14
   7     58      506
  15    242     4060      65512
  31    994    32618    1048336    33554312

Links

Crossrefs

Cf. A000312, A002416, A036679, A101030, A131689, A199656, A344110, A344112, A344113, A344115.

Programs

  • Mathematica
    TableForm[Table[2^(n*k) - Sum[Binomial[k, k - i] (k - i)^n*(-1)^i, {i, 0, k}], {n, 5}, {k, n}]]
  • PARI
    T(n,k) = 2^(n*k) - k!*stirling(n, k, 2); \\ Michel Marcus, Jun 26 2021

Formula

T(n,k) = 2^(n*k) - k!*Stirling2(n,k).
T(n,k) = A344110(n,k) - A131689(n,k).

A344260 a(n) is the number of relations from an n-element set into a set of at most n elements.

Original entry on oeis.org

1, 3, 21, 585, 69905, 34636833, 69810262081, 567382630219905, 18519084246547628289, 2422583247133816584929793, 1268889750375080065623288448001, 2659754699919401766201267083003561985, 22306191045953951743035482794815064402563073, 748380193317489370459454048174977015562807531282433
Offset: 0

Views

Author

Stefano Spezia, May 13 2021

Keywords

Comments

Symmetrically, also the number of relations from a set of at most n elements into an n-element set.

Crossrefs

Row sums of A344110.
Cf. A000079, A002416, A031971, A089072, A117401, A275779.

Programs

  • Mathematica
    Join[{1},Table[(2^(n+n^2)-1)/(2^n-1),{n,13}]]

Formula

a(n) = (2^(n+n^2) - 1)/(2^n - 1) for n > 0 and a(0) = 1.
a(n) ~ 2^(n^2).
a(n) = A275779(n) + 1. - Hugo Pfoertner, May 14 2021

A347034 Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.

Original entry on oeis.org

0, 0, 2, 0, 3, 21, 0, 4, 40, 232, 0, 5, 65, 505, 3005, 0, 6, 96, 936, 7056, 45936, 0, 7, 133, 1561, 14287, 112609, 818503, 0, 8, 176, 2416, 26048, 241984, 2056832, 16736896, 0, 9, 225, 3537, 43929, 470961, 4601529, 42683841, 387057609, 0, 10, 280, 4960, 69760, 848800
Offset: 1

Views

Author

Mohammad K. Azarian, Aug 28 2021

Keywords

Comments

The formula for this sequence is Theorem 2.2(iv) of the author's paper, p. 131 (see the link).

Examples

			For T(2,3): the number of functions is 3^2 and the number of one-to-one functions is 6, so 3^2 - 6 = 3 and thus T(2,3) = 3.
Triangle T(n,k) begins:
       k=1  k=2   k=3   k=4    k=5     k=6
  n=1:  0    0    0     0      0       0
  n=2:       2    3     4      5       6
  n=3:            21    40     65      96
  n=4:                  232    505     936
  n=5:                         3005    7056
  n=6:                                 45936

Links

Crossrefs

Cf. A000312, A002416, A036679, A068424, A089072, A101030, A199656, A344110, A344112, A344113, A344114, A344115, A344116.

Programs

  • Maple
    A347034 := proc(n,k)
    k^n-k!/(k-n)! ;
end proc:
seq(seq(A347034(n,k),n=1..k),k=1..12) ; # R. J. Mathar, Jan 12 2023
  • Mathematica
    Table[k^n - k!/(k - n)!, {k, 12}, {n, k}] // Flatten
  • PARI
    T(n,k) = k^n - k!/(k - n)!;
row(k) = vector(k, i, T(i, k)); \\ Michel Marcus, Oct 01 2021

Formula

T(n,k) = k^n - k!/(k - n)!, k>=n.
T(n,n) = A036679(n).
Showing 1-7 of 7 results.

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