A245348 -id:A245348 - cp's OEIS frontend

A089945 Main diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.

1, 3, 15, 112, 1125, 14256, 218491, 3932160, 81310473, 1900000000, 49516901511, 1424099377152, 44804009850925, 1530735634132992, 56439656982421875, 2233785415175766016, 94459960699823921169, 4250383588380798812160, 202774313738037680879743

Offset: 0

Paul D. Hanna, Nov 23 2003

nonn

a(n) is the number of labeled trees on n+1 nodes with a designated node or edge. - Geoffrey Critzer, Mar 25 2017

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A089944, A089946, A245348.

[(2*n+1)*(n+1)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 16 2017

nn = 30; T[z_] = -LambertW[-z]; Drop[Range[0, nn]! CoefficientList[Series[T[z] + T[z]^2/2, {z, 0, nn}], z], 1] (* Geoffrey Critzer, Mar 25 2017 *)
Table[(2 n + 1) (n + 1)^(n - 1), {n, 0, 18}] (* Michael De Vlieger, Mar 25 2017 *)

PARI
```
a(n)=if(n<0,0,(2*n+1)*(n+1)^(n-1))
```

a(n) = (2*n+1)*(n+1)^(n-1).

E.g.f.: (-LambertW(-x)/x)*(1-LambertW(-x))/(1+LambertW(-x)).

A239771 Number of pairs of functions (f,g) from a size n set into itself satisfying f(x) = g(g(f(x))).

Original entry on oeis.org

1, 1, 10, 213, 8056, 465945, 37823616, 4075467781, 560230714240, 95369455852497, 19643693349548800, 4805295720474420501, 1374890520609054683136, 454286686896040037996905, 171479277693049020232695808, 73262491601904459123264721125, 35143072854722729593790081499136

Offset: 0

Chad Brewbaker, Mar 26 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A000085, A048993, A181162, A239769, A239773, A245348, A241015.

Column k=2 of A245980.

g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
a:= n-> add(binomial(n, k)*Stirling2(n, k)*k!*
        add(binomial(n-k, i)*binomial(k, i)*i!*
        g(k-i)*n^(n-k-i), i=0..min(k, n-k)), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 18 2014

g[n_] := g[n] = If[n < 2, 1, g[n-1] + (n-1)*g[n-2]];
a[n_] := If[n == 0, 1, Sum[Binomial[n, k]*StirlingS2[n, k]*k!*Sum[ Binomial[n-k, i]*Binomial[k, i]*i!*g[k-i]*n^(n-k-i), {i, 0, Min[k, n-k]} ], {k, 0, n}]];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

a(n) = Sum_{k=0..n} C(n,k) * A048993(n,k) * k! * A245348(n,k). - Alois P. Heinz, Jul 18 2014

a(6)-a(7) from Giovanni Resta, Mar 28 2014

a(8)-a(16) from Alois P. Heinz, Jul 18 2014

A245692 Number T(n,k) of endofunctions f on [n] that are self-inverse on [k] but not on [k+1]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 12, 7, 4, 4, 144, 62, 28, 12, 10, 2000, 695, 264, 100, 40, 26, 32400, 9504, 3126, 1050, 370, 130, 76, 605052, 154007, 44716, 13458, 4256, 1366, 456, 232, 12845056, 2891776, 751872, 204776, 58784, 17292, 5272, 1624, 764

Offset: 0

Alois P. Heinz, Jul 29 2014

T(n,k) counts endofunctions f:{1,...,n}-> {1,...,n} with f(f(i))=i for all i in {1,...,k} and f(f(k+1))<>k+1 if k

			T(3,1) = 7: (1,1,1), (1,1,2), (1,1,3), (1,3,1), (1,3,3), (3,1,1), (3,3,1).
T(3,2) = 4: (1,2,1), (1,2,2), (2,1,1), (2,1,2).
T(3,3) = 4: (1,2,3), (1,3,2), (2,1,3), (3,2,1).
Triangle T(n,k) begins:
0 :       1;
1 :       0,      1;
2 :       1,      1,     2;
3 :      12,      7,     4,     4;
4 :     144,     62,    28,    12,   10;
5 :    2000,    695,   264,   100,   40,   26;
6 :   32400,   9504,  3126,  1050,  370,  130,  76;
7 :  605052, 154007, 44716, 13458, 4256, 1366, 456, 232;
     ...

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

Column k=0 gives A076728 for n>1.
Row sums give A000312.
Main diagonal gives A000085.
Cf. A245348, A245693 (the same for permutations).

g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
H:= (n, k)-> add(binomial(n-k, i)*binomial(k, i)*i!*
             g(k-i)*n^(n-k-i), i=0..min(k, n-k)):
T:= (n, k)-> H(n, k) -H(n, k+1):
seq(seq(T(n, k), k=0..n), n=0..10);

g[n_] := g[n] = If[n<2, 1, g[n-1] + (n-1)*g[n-2]]; H[0, 0] = 1; H[n_, k_] := Sum[Binomial[n-k, i]*Binomial[k, i]*i!*g[k-i]*n^(n-k-i), {i, 0, Min[k, n-k]}]; T[n_, k_] := H[n, k] - H[n, k+1]; Table[T[n, k], {n, 0, 10}, { k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

T(n,k) = A245348(n,k) - A245348(n,k+1).

A245141 Number of endofunctions f on [2n] that are self-inverse on [n].

Original entry on oeis.org

1, 3, 50, 1626, 83736, 6026120, 571350096, 67996818960, 9862902275456, 1700092943088768, 342087177215788800, 79115601821198404352, 20779757607847901690880, 6133520505473954148381696, 2017134796016735182500521984, 733523863838078950241395968000

Offset: 0

Alois P. Heinz, Jul 21 2014

nonn

a(n) counts endofunctions f:{1,...,2n}-> {1,...,2n} with f(f(i))=i for all i in {1,...,n}.

			a(1) = 3: (1,1), (1,2), (2,1).

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000085, A245348.

Column k=2 of A246070.

g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
a:= n-> add(binomial(n, i)^2*i!*g(n-i)*(2*n)^(n-i), i=0..n):
seq(a(n), n=0..20);

Join[{1}, Table[n! * Sum[Binomial[n,k] * 2^k * n^k* Sum[1/((k - 2*j)!*2^j*j!), {j, 0, Floor[k/2]}], {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 05 2021 *)

a(n) = Sum_{i=0..n} C(n,i)^2 * i! * A000085(n-i) * (2*n)^(n-i).

a(n) = A245348(2n,n).

Showing 1-4 of 4 results.

cp's OEIS Frontend

A089945 Main diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A239771 Number of pairs of functions (f,g) from a size n set into itself satisfying f(x) = g(g(f(x))).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A245692 Number T(n,k) of endofunctions f on [n] that are self-inverse on [k] but not on [k+1]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A245141 Number of endofunctions f on [2n] that are self-inverse on [n].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula