A243203 -id:A243203 - cp's OEIS frontend

A244137 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

1, 0, 1, 0, 2, 2, 0, 12, 6, 9, 0, 108, 48, 36, 64, 0, 1280, 540, 360, 320, 625, 0, 18750, 7680, 4860, 3840, 3750, 7776, 0, 326592, 131250, 80640, 60480, 52500, 54432, 117649, 0, 6588344, 2612736, 1575000, 1146880, 945000, 870912, 941192, 2097152

Offset: 0

Stanislav Sykora, Jun 22 2014

T(n,k)=(k)^(k-1)*(n-k)^(n-k)*binomial(n,k) for k>0, while T(n,0)=0^n by convention.

There are many binomial decompositions of n^n, some with all terms positive like this one (see A243203). However, for every n, the terms corresponding to k=1..n in this one are exceptionally similar in value (at least on log scale).

			First rows of the triangle, all summing up to n^n:
1,
0, 1,
0, 2, 2,
0, 12, 6, 9,
0, 108, 48, 36, 64,
0, 1280, 540, 360, 320, 625,

Stanislav Sykora, Table of n, a(n) for rows 0..100
S. Sykora, An Abel's Identity and its Corollaries, Stan's Library, Volume V, 2014, DOI 10.3247/SL5Math14.004. See eq.(13), with b=-1.

Cf. A243203, A244116, A244117, A244118, A244119, A244120, A244121, A244122, A244123, A244124, A244125, A244126, A244127, A244128, A244129, A244130, A244131, A244132, A244133, A244134, A244135, A244136, A244138, A244139, A244140, A244141, A244142, A244143.

seq(nmax, b)={my(v, n, k, irow);
v = vector((nmax+1)*(nmax+2)/2); v[1]=1;
for(n=1, nmax, irow=1+n*(n+1)/2; v[irow]=0;
  for(k=1, n, v[irow+k]=(-k*b)^(k-1)*(n+k*b)^(n-k)*binomial(n, k); ); );
return(v); }
a=seq(100,-1);

A066324 Number of endofunctions on n labeled points constructed from k rooted trees.

Original entry on oeis.org

1, 2, 2, 9, 12, 6, 64, 96, 72, 24, 625, 1000, 900, 480, 120, 7776, 12960, 12960, 8640, 3600, 720, 117649, 201684, 216090, 164640, 88200, 30240, 5040, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320, 43046721

Offset: 1

Christian G. Bower, Dec 14 2001

T(n,k) = number of endofunctions with k recurrent elements. - Mitch Harris, Jul 06 2006

The sum of row n is n^n, for any n. Basically the same sequence arises when studying random mappings (see A243203, A243202). - Stanislav Sykora, Jun 01 2014

			Triangle T(n,k) begins:
       1;
       2,      2;
       9,     12,      6;
      64,     96,     72,     24;
     625,   1000,    900,    480,   120;
    7776,  12960,  12960,   8640,  3600,   720;
  117649, 201684, 216090, 164640, 88200, 30240, 5040;
  ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 87, see (2.3.28).
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.32.

Alois P. Heinz, Rows n = 1..141, flattened

Column 1: A000169.
Main diagonal: A000142.
T(n, n-1): A062119.
Row sums give A000312.
Cf. A001864, A021010, A063169, A122525, A144084, A243203.

T:= (n, k)-> k*n^(n-k)*(n-1)!/(n-k)!:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 22 2012

f[list_] := Select[list, # > 0 &]; t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Flatten[Map[f, Drop[Range[0, 10]! CoefficientList[Series[1/(1 - y*t), {x, 0, 10}], {x, y}], 1]]] (* Geoffrey Critzer, Dec 05 2011 *)

T(n, k)=k*n^(n-k)*(n-1)!/(n-k)! \\ Charles R Greathouse IV, Dec 05 2011

T(n,k) = k*n^(n-k)*(n-1)!/(n-k)!.
E.g.f. (relative to x): A(x, y)=1/(1-y*B(x)) - 1 = y*x +(2*y+2*y^2)*x^2/2! + (9*y+12*y^2+6*y^3)*x^3/3! + ..., where B(x) is e.g.f. A000169.
From Peter Bala, Sep 30 2011: (Start)
Let F(x,t) = x/(1+t*x)*exp(-x/(1+t*x)) = x*(1 - (1+t)*x + (1+4*t+2*t^2)*x^2/2! - ...). F is essentially the e.g.f. for A144084 (see also A021010). Then the e.g.f. for the present table is t*F(x,t)^(-1), where the compositional inverse is taken with respect to x.
Removing a factor of n from the n-th row entries results in A122525 in row reversed form.
(End)
Sum_{k=2..n} (k-1) * T(n,k) = A001864(n). - Geoffrey Critzer, Aug 19 2013
Sum_{k=1..n} k * T(n,k) = A063169(n). - Alois P. Heinz, Dec 15 2021

A243202 Coefficients of a particular decomposition of N^N in terms of binomial coefficients.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 3, 4, 6, 0, 16, 16, 18, 24, 0, 125, 100, 90, 96, 120, 0, 1296, 864, 648, 576, 600, 720, 0, 16807, 9604, 6174, 4704, 4200, 4320, 5040, 0, 262144, 131072, 73728, 49152, 38400, 34560, 35280, 40320, 0

Offset: 0

Stanislav Sykora, Jun 01 2014

a(n) is an element in the triangle of coefficients c(N,j), N = 0,1,2,3,... denoting a row, and j = 0,1,2,...r, specified numerically by the formula below. For any row N, Sum(j=0..N)(c(N,j)*binomial(N,j)) = N^N. Note that all rows start with 0, which makes them easily recognizable. It is believed that keeping the zero terms is preferable because it makes the summation run over all admissible j's in the binomial.

			The first rows of the triangle are (first item is the row number N):
0 0
1 0, 1
2 0, 1, 2
3 0, 3, 4, 6
4 0, 16, 16, 18, 24
5 0, 125, 100, 90, 96, 120
6 0, 1296, 864, 648, 576, 600, 720
7 0, 16807, 9604, 6174, 4704, 4200, 4320, 5040
8 0, 262144, 131072, 73728, 49152, 38400, 34560, 35280, 40320

Stanislav Sykora, Table of n, a(n) for rows 0..100, flattened
S. Sykora, A Random Mapping Statistics and a Related Identity, Stan's Library, Volume V, June 2014.

Cf. A243203.

A243202(maxrow) = {
  my(v,n,j,irow,f);v = vector((maxrow+1)*(maxrow+2)/2);
  for(n=1,maxrow,irow=1+n*(n+1)/2;v[irow]=0;f=1;
  for(j=1,n,f *= j;v[irow+j] = j*f*n^(n-j-1);););
  return(v);}

c(N,j)=N^(N-j)*(j/N)*j! for N>0 and 0<=j<=N, and c(N,j)=0 otherwise.

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A244137 Triangle read by rows: terms T(n,k) of a binomial decomposition of n^n as Sum(k=0..n)T(n,k).

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A066324 Number of endofunctions on n labeled points constructed from k rooted trees.

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A243202 Coefficients of a particular decomposition of N^N in terms of binomial coefficients.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

PARI

Formula