A089945 -id:A089945 - cp's OEIS frontend

A089946 Secondary diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.

1, 4, 24, 200, 2160, 28812, 458752, 8503056, 180000000, 4287177620, 113515167744, 3308603804376, 105288694411264, 3632897460937500, 135107988821114880, 5388090449900829728, 229385780960233586688, 10383890888434362036516, 498073600000000000000000

Offset: 0

Paul D. Hanna, Nov 23 2003

nonn

Also the hyperbinomial transform of A089945 (the main diagonal of A089944): a(n) = Sum_{k=0..n} C(n,k)*(n-k+1)^(n-k-1)*A089945(k).
With offset 1, a(n) = total number of children of the root in all (n+1)^(n-1) trees on {0,1,2,...,n} rooted at 0. For example, with edges directed away from the root, the trees on {0,1,2} are {0->1,0->2},{0->1->2},{0->2->1} and contain a total of a(2)=4 children of 0. - David Callan, Feb 01 2007
With offset 1, a(n) is the number of labeled rooted trees in all rooted forests on n nodes. The E.g.f. is B(T(x)) where  B(x)=x*exp(x) and T(x) is Euler's tree function. - Geoffrey Critzer, Oct 07 2011

Alois P. Heinz, Table of n, a(n) for n = 0..386
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
F. A. Haight, Letter to N. J. A. Sloane, n.d.

Cf. A089944, A089945.

A diagonal of A259334.

[2*(n+1) * (n+2)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 14 2017

t=Sum[n^(n-1)x^n/n!, {n,1,20}]; Drop[Range[0,20]!*CoefficientList[ Series[t*Exp[t], {x,0,20}], x], 1] (* Geoffrey Critzer, Oct 07 2011 *)
Table[2*(n+1)*(n+2)^(n-1), {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)

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

a(n) = 2*(n+1) * (n+2)^(n-1).
a(n) = Sum_{k=0..n} C(n, k) * (n-k+1)^(n-k-1) * (2*k+1) * (k+1)^(k-1).
E.g.f.: (-LambertW(x)/x)^2 * (1 - LambertW(x)) / (1 + LambertW(x)).

A245348 Number T(n,k) of endofunctions f on [n] that are self-inverse on [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 4, 3, 2, 27, 15, 8, 4, 256, 112, 50, 22, 10, 3125, 1125, 430, 166, 66, 26, 46656, 14256, 4752, 1626, 576, 206, 76, 823543, 218491, 64484, 19768, 6310, 2054, 688, 232, 16777216, 3932160, 1040384, 288512, 83736, 24952, 7660, 2388, 764

Offset: 0

Alois P. Heinz, Jul 18 2014

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

			T(3,1) = 15: (1,1,1), (2,1,1), (3,1,1), (1,2,1), (3,2,1), (1,3,1), (3,3,1), (1,1,2), (2,1,2), (1,2,2), (1,3,2), (1,1,3), (2,1,3), (1,2,3), (1,3,3).
T(3,2) = 8: (2,1,1), (1,2,1), (3,2,1), (2,1,2), (1,2,2), (1,3,2), (2,1,3), (1,2,3).
T(3,3) = 4: (3,2,1), (1,3,2), (2,1,3), (1,2,3).
Triangle T(n,k) begins:
0 :       1;
1 :       1,      1;
2 :       4,      3,     2;
3 :      27,     15,     8,     4;
4 :     256,    112,    50,    22,   10;
5 :    3125,   1125,   430,   166,   66,   26;
6 :   46656,  14256,  4752,  1626,  576,  206,  76;
7 :  823543, 218491, 64484, 19768, 6310, 2054, 688, 232;
     ...

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

Columns k=0-1 give: A000312, A089945(n-1) for n>0.
Main diagonal gives A000085.
T(2n,n) gives A245141.
Cf. A239771, A245692.

g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
T:= (n, k)-> add(binomial(n-k, i)*binomial(k, i)*i!*
             g(k-i)*n^(n-k-i), i=0..min(k, n-k)):
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]]; T[0, 0] = 1; T[n_, k_] := Sum[Binomial[n-k, i]*Binomial[k, i]*i!*g[k-i]*n^(n-k-i), {i, 0, Min[k, n-k]}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

T(n,k) = Sum_{i=0..min(k,n-k)} C(n-k,i)*C(k,i)*i!*A000085(k-i)*n^(n-k-i).

A089944 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 8, 4, 1, 5, 20, 15, 5, 1, 6, 48, 54, 24, 6, 1, 7, 112, 189, 112, 35, 7, 1, 8, 256, 648, 512, 200, 48, 8, 1, 9, 576, 2187, 2304, 1125, 324, 63, 9, 1, 10, 1280, 7290, 10240, 6250, 2160, 490, 80, 10, 1, 11, 2816, 24057, 45056, 34375, 14256, 3773, 704, 99, 11, 1

Offset: 0

Paul D. Hanna, Nov 23 2003

The main diagonal is A089945: {T(n,n)=(2*n+1)*(n+1)^(n-1), n>=0}; the hyperbinomial transform of the main diagonal is the next lower diagonal in the array (A089946): {T(n+1,n) = 2*(n+1)*(n+2)^(n-1), n>=0}.

			Rows begin:
  {1, 2, 3, 4, 5, 6, 7,..},
  {1, 3, 8, 20, 48, 112, 256,..},
  {1, 4, 15, 54, 189, 648, 2187,..},
  {1, 5, 24, 112, 512, 2304, 10240,..},
  {1, 6, 35, 200, 1125, 6250, 34375,..},
  {1, 7, 48, 324, 2160, 14256, 93312,..},
  {1, 8, 63, 490, 3773, 28812, 218491,..},..

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals).

Cf. A089945, A089946.
Rows : A000027, A001792, A006234, A079028, A081105, A081106, A081107, A081108, A081109, A081122.
Columns : A000012, A000027, A005563.

A089944[n_, k_] := (k + n + 1)*(n + 1)^(k - 1);
Table[A089944[k, n - k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2025 *)

T(n,k)=if(n<0 || k<0,0,(k+n+1)*(n+1)^(k-1))

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

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

cp's OEIS Frontend

A089946 Secondary diagonal of array A089944, in which the n-th row is the n-th binomial transform of the natural numbers.

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A245348 Number T(n,k) of endofunctions f on [n] that are self-inverse on [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A089944 Square array, read by antidiagonals, where the n-th row is the n-th binomial transform of the natural numbers, with T(0,k) = (k+1) for k>=0.

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Mathematica

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