A217701 -id:A217701 - cp's OEIS frontend

A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 9, 1, 4, 10, 16, 24, 44, 1, 5, 17, 38, 65, 120, 265, 1, 6, 26, 78, 168, 326, 720, 1854, 1, 7, 37, 142, 393, 872, 1957, 5040, 14833, 1, 8, 50, 236, 824, 2208, 5296, 13700, 40320, 133496, 1, 9, 65, 366, 1569, 5144, 13977, 37200, 109601, 362880, 1334961

Offset: 0

Philippe Deléham, Dec 12 2003

Can be extended to columns with negative indices k<0 via T(n,k) = A292977(n,-k). - Max Alekseyev, Mar 06 2018

			n\k -1   0   1    2    3    4     5     6  ...
----------------------------------------------
0  | 1,  1,  1,   1,   1,   1,    1,    1, ...
1  | 0,  1,  2,   3,   4,   5,    6,    7, ...
2  | 1,  2,  5,  10,  17,  26,   37,   50, ...
3  | 2,  6, 16,  38,  78, 152,  236,  366, ...
4  | 9, 24, 65, 168, 393, 824, 1569, 2760, ...
...

Columns: A000166, A000142, A000522, A010842, A053486, A053487, A080954.
Main diagonal gives A217701.
Cf. A080955, A008290, A292977.

(* Assuming offset (0, 0): *)
T[n_, k_] := Exp[k - 1] Gamma[n + 1, k - 1];
Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten  (* Peter Luschny, Dec 24 2021 *)

For n > 0, k >= -1, T(n,k) is the permanent of the n X n matrix with k+1 on the diagonal and 1 elsewhere.
T(0,k) = 1.
T(n,k) = Sum_{j>=0} A008290(n,j) * (k+1)^j.
T(n,k) = n*T(n-1, k) + k^n .
T(n,k) = n! * Sum_{j=0..n} k^j/j!.
E.g.f. for k-th column: exp(k*x)/(1-x).
Assuming n >= 0, k >= 0: T(n, k) = exp(k-1)*Gamma(n+1, k-1). - Peter Luschny, Dec 24 2021

Edited and changed offset for k to -1 by Max Alekseyev, Mar 08 2018

A349454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 8, 3, 0, 1, 81, 32, 6, 0, 1, 1024, 405, 80, 10, 0, 1, 15625, 6144, 1215, 160, 15, 0, 1, 279936, 109375, 21504, 2835, 280, 21, 0, 1, 5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1, 134217728, 51883209, 10077696, 1312500, 129024, 10206, 672, 36, 0, 1

Offset: 0

Alois P. Heinz, Dec 30 2021

			Triangle T(n,k) begins:
        1;
        0,       1;
        1,       0,      1;
        8,       3,      0,     1;
       81,      32,      6,     0,    1;
     1024,     405,     80,    10,    0,   1;
    15625,    6144,   1215,   160,   15,   0,  1;
   279936,  109375,  21504,  2835,  280,  21,  0, 1;
  5764801, 2239488, 437500, 57344, 5670, 448, 28, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Column k=0 gives A065440.
Row sums give A204042.
Main diagonal and first lower diagonal give A000012, A000004.
T(n+1,n-1) gives A000217.
T(n+3,n) gives A130809.
T(n+3,n-1) gives A102741 for n>=1.
Cf. A055134, A217701, A350212, A350454.

T:= (n, k)-> binomial(n, k)*(n-k-1)^(n-k):
seq(seq(T(n, k), k=0..n), n=0..10);

T(n,k) = binomial(n,k) * (n-k-1)^(n-k).
From Mélika Tebni, Apr 02 2023: (Start)
E.g.f. of column k: -x / (LambertW(-x)*(1+LambertW(-x)))*x^k / k!.
Sum_{k=0..n} k^k*T(n,k) = A217701(n). (End)

A350297 Triangle read by rows: T(n,k) = n!*(n-1)^k/k!.

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 6, 12, 12, 8, 24, 72, 108, 108, 81, 120, 480, 960, 1280, 1280, 1024, 720, 3600, 9000, 15000, 18750, 18750, 15625, 5040, 30240, 90720, 181440, 272160, 326592, 326592, 279936, 40320, 282240, 987840, 2304960, 4033680, 5647152, 6588344, 6588344, 5764801

Offset: 0

Robert B Fowler, Dec 23 2021

Rows n >= 2 are coefficients in a double summation power series for the integral of x^(1/x), and the integral of its inverse function y^(y^(y^(y^(...)))). See A350358.

			Triangle T(n,k) begins:
  -----------------------------------------------------------------
   n\k     0      1      2       3       4       5       6       7
  -----------------------------------------------------------------
   0  |    1,
   1  |    1,     0,
   2  |    2,     2,     1,
   3  |    6,    12,    12,      8,
   4  |   24,    72,   108,    108,     81,
   5  |  120,   480,   960,   1280,   1280,   1024,
   6  |  720,  3600,  9000,  15000,  18750,  18750,  15625,
   7  | 5040, 30240, 90720, 181440, 272160, 326592, 326592, 279936.
  ...

Cf. A000142 (first column), A062119 (second column), A065440 (main diagonal), A055897 (subdiagonal), A217701 (row sums).
Cf. A350269, A350358.
Cf. A061711, A350149.

T := (n, k) -> (n!/k!)*(n - 1)^k:
seq(seq(T(n, k), k = 0..n), n = 0..8); # Peter Luschny, Dec 24 2021

T[1, 0] := 1; T[n_, k_] := n!*(n - 1)^k/k!; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 24 2021 *)

T(n, k) = binomial(n, k)*A350269(n, k). - Peter Luschny, Dec 25 2021

T(n+1, k) = A061711(n) * (n+1) / A350149(n, k). - Robert B Fowler, Jan 11 2022

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

Alois P. Heinz, Dec 31 2021

			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;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

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.

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

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 *)

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)

cp's OEIS Frontend

A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A349454 Number T(n,k) of endofunctions on [n] with exactly k fixed points, all of which are isolated; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A350297 Triangle read by rows: T(n,k) = n!*(n-1)^k/k!.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula