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 1-4 of 4 results.

A350212 Number T(n,k) of endofunctions on [n] with exactly k isolated fixed points; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 17, 9, 0, 1, 169, 68, 18, 0, 1, 2079, 845, 170, 30, 0, 1, 31261, 12474, 2535, 340, 45, 0, 1, 554483, 218827, 43659, 5915, 595, 63, 0, 1, 11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1, 262517615, 102030777, 19961388, 2625924, 261954, 21294, 1428, 108, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Dec 19 2021

Keywords

Examples

			T(3,1) = 9: 122, 133, 132, 121, 323, 321, 113, 223, 213.
Triangle T(n,k) begins:
         1;
         0,       1;
         3,       0,      1;
        17,       9,      0,      1;
       169,      68,     18,      0,     1;
      2079,     845,    170,     30,     0,   1;
     31261,   12474,   2535,    340,    45,   0,  1;
    554483,  218827,  43659,   5915,   595,  63,  0, 1;
  11336753, 4435864, 875308, 116424, 11830, 952, 84, 0, 1;
  ...

Links

Crossrefs

Columns k=0-1 give: |A069856|, A348590.
Row sums give A000312.
T(n+1,n-1) gives A045943.
Cf. A001865, A008290, A008291, A055134, A055897, A060281, A086659, A124323, A350134, A349454.

Programs

  • Maple
    g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end:
b:= proc(n, m) option remember; `if`(n=0, x^m, add(g(i)*
      b(n-i, m+`if`(i=1, 1, 0))*binomial(n-1, i-1), i=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);
# second Maple program:
A350212 := (n,k)-> add((-1)^(j-k)*binomial(j,k)*binomial(n,j)*(n-j)^(n-j), j=0..n):
seq(print(seq(A350212(n, k), k=0..n)), n=0..9); # Mélika Tebni, Nov 24 2022
  • Mathematica
    g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}];
b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[g[i]*
     b[n - i, m + If[i == 1, 1, 0]]*Binomial[n - 1, i - 1], {i, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 11 2022, after Alois P. Heinz *)

Formula

Sum_{k=0..n} k * T(n,k) = A055897(n).
Sum_{k=1..n} T(n,k) = A350134(n).
From Mélika Tebni, Nov 24 2022: (Start)
T(n,k) = binomial(n, k)*|A069856(n-k)|.
E.g.f. column k: exp(-x)*x^k / ((1 + LambertW(-x))*k!).
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*binomial(j, k)*binomial(n, j)*(n-j)^(n-j). (End)

A204042 The number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that all of the fixed points in f are isolated.

Original entry on oeis.org

1, 1, 2, 12, 120, 1520, 23160, 413952, 8505280, 197631072, 5125527360, 146787894440, 4601174623584, 156693888150384, 5761055539858528, 227438694372072120, 9596077520725211520, 430920897407809702208, 20520683482765477749120, 1032920864149903149579336, 54797532208320308334631840
Offset: 0

Views

Author

Geoffrey Critzer, Jan 09 2012

Keywords

Comments

Note this sequence counts the functions enumerated by A065440 for which the statement is vacuously true.
a(n) is also the number of partial endofunctions on {1,2,...,n} without fixed points.

Examples

			a(2)=2 because there are two functions f:{1,2}->{1,2} in which all the fixed points are isolated: 1->1,2->2  and 1->2,2->1 (which has no fixed points).

Links

Crossrefs

Cf. A065440, A086331, A350134.
Row sums of A349454.

Programs

  • Maple
    a:= n-> add((j-1)^j*binomial(n, j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 16 2021
  • Mathematica
    t = Sum[n^(n-1) x^n/n!, {n,1,20}]; Range[0,20]! CoefficientList[Series[Exp[x] Exp[Log[1/(1-t)]-t], {x,0,20}], x]

Formula

E.g.f.: exp(x)*A(x) where A(x) is the e.g.f. for A065440.
a(n) ~ exp(exp(-1)-1)*n^n. - Vaclav Kotesovec, Sep 24 2013
a(n) = Sum_{j=0..n} (j-1)^j * binomial(n,j). - Alois P. Heinz, Dec 16 2021

A350454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, none of which are isolated; triangle T(n,k), n >= 0, 0 <= k <= n/2, read by rows.

Original entry on oeis.org

1, 0, 1, 2, 8, 9, 81, 76, 12, 1024, 875, 180, 15625, 12606, 2910, 120, 279936, 217217, 53550, 3780, 5764801, 4348856, 1118936, 102480, 1680, 134217728, 99111735, 26280072, 2817360, 90720, 3486784401, 2532027610, 686569050, 81864720, 3729600, 30240
Offset: 0

Views

Author

Alois P. Heinz, Dec 31 2021

Keywords

Examples

			Triangle T(n,k) begins:
           1;
           0;
           1,          2;
           8,          9;
          81,         76,        12;
        1024,        875,       180;
       15625,      12606,      2910,      120;
      279936,     217217,     53550,     3780;
     5764801,    4348856,   1118936,   102480,    1680;
   134217728,   99111735,  26280072,  2817360,   90720;
  3486784401, 2532027610, 686569050, 81864720, 3729600, 30240;
  ...

Links

Crossrefs

Column k=0 gives A065440.
Row sums give |A069856|.
T(2n,n) gives A001813.
Cf. A349454.

Programs

  • Maple
    c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
t:= proc(n) option remember; n^(n-1) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*
      binomial(n-1, i-1)*(c(i)+`if`(i=1, 0, x*t(i))), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);
# second Maple program:
egf := k-> exp(LambertW(-x))*(-x-LambertW(-x))^k/((1+LambertW(-x))*k!):
A350454 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A350454(n, k), k=0..n/2)), n=0..9); # Mélika Tebni, Nov 22 2022
  • Mathematica
    c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
t[n_] := t[n] = n^(n - 1);
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - i]*
     Binomial[n - 1, i - 1]*(c[i] + If[i == 1, 0, x*t[i]]), {i, 1, n}]]];
T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 06 2022, after Alois P. Heinz *)

Formula

E.g.f. column k: exp(W(-x))*(-x - W(-x))^k / ((1 + W(-x))*k!), W(x) the Lambert W-function. - Mélika Tebni, Nov 22 2022
From Mélika Tebni, Dec 22 2022: (Start)
For n > 1, T(n,1) = n*A045531(n-1).
Sum_{k=0..n} (-1)^(n-k)*T(n+k,k) = 2^n.
Sum_{k=0..n} (-1)^(n-k)*T(n+k,k)/(n+k-1) = 1/n, for n > 1. (End)

A350446 Number T(n,k) of endofunctions on [n] with exactly k cycles of length larger than 1; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 3, 1, 16, 11, 125, 128, 3, 1296, 1734, 95, 16807, 27409, 2425, 15, 262144, 499400, 61054, 945, 4782969, 10346328, 1605534, 42280, 105, 100000000, 240722160, 44981292, 1706012, 11025, 2357947691, 6222652233, 1351343346, 67291910, 763875, 945
Offset: 0

Views

Author

Alois P. Heinz, Dec 31 2021

Keywords

Examples

			Triangle T(n,k) begins:
           1;
           1;
           3,          1;
          16,         11;
         125,        128,          3;
        1296,       1734,         95;
       16807,      27409,       2425,       15;
      262144,     499400,      61054,      945;
     4782969,   10346328,    1605534,    42280,    105;
   100000000,  240722160,   44981292,  1706012,  11025;
  2357947691, 6222652233, 1351343346, 67291910, 763875, 945;
  ...

Links

Crossrefs

Column k=0 gives A000272(n+1).
Row sums give A000312.
T(2n,n) gives A001147.
Cf. A055134, A060281, A349454, A350452.
Cf. A136394, A190314, A217701.

Programs

  • Maple
    c:= proc(n) option remember; add(n!*n^(n-k-1)/(n-k)!, k=2..n) end:
t:= proc(n) option remember; n^(n-1) end:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
      b(n-i)*binomial(n-1, i-1)*(c(i)*x+t(i)), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);
# second Maple program:
egf := k-> (LambertW(-x)-log(1+LambertW(-x)))^k/(exp(LambertW(-x))*k!):
A350446 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A350446(n, k), k=0..n/2)), n=0..10); # Mélika Tebni, Mar 23 2023
  • Mathematica
    c[n_] := c[n] = Sum[n!*n^(n - k - 1)/(n - k)!, {k, 2, n}];
t[n_] := t[n] = n^(n - 1);
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[
     b[n - i]*Binomial[n - 1, i - 1]*(c[i]*x + t[i]), {i, 1, n}]]];
T[n_] :=  With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 06 2022, after Alois P. Heinz *)

Formula

From Mélika Tebni, Mar 23 2023: (Start)
E.g.f. of column k: (W(-x)-log(1 + W(-x)))^k / (exp(W(-x))*k!), W(x) the Lambert W-function.
T(n,k) = Sum_{j=k..n} n^(n-j)*binomial(n-1,j-1)*A136394(j,k), for n > 0.
T(n,k) = Sum_{j=k..n} (n-j+1)^(n-j-1)*binomial(n,j)*A350452(j,k).
Sum_{k=0..n/2} (k+1)*T(n,k) = A190314(n), for n > 0.
Sum_{k=0..n/2} 2^k*T(n,k) = A217701(n). (End)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.