A243203 -id:A243203

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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).

+10
28

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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).

LINKS

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.

EXAMPLE

First rows of the triangle, all summing up to n^n:

0, 1,

0, 2, 2,

0, 12, 6, 9,

0, 108, 48, 36, 64,

0, 1280, 540, 360, 320, 625,

PROG

(PARI) 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);

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

Stanislav Sykora, Jun 22 2014

STATUS

approved

A066324

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

+10
7

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

REFERENCES

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.

LINKS

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

FORMULA

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

EXAMPLE

Triangle T(n,k) begins:

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;

...

MAPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Column 1: A000169.

Main diagonal: A000142.

T(n, n-1): A062119.

Row sums give A000312.

Cf. A001864, A021010, A063169, A122525, A144084, A243203.

KEYWORD

nonn,tabl,changed

AUTHOR

Christian G. Bower, Dec 14 2001

STATUS

approved

A243202

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

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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.

LINKS

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.

FORMULA

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

EXAMPLE

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

PROG

(PARI) 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); }

CROSSREFS

Cf. A243203.

KEYWORD

nonn,easy,tabl

AUTHOR

Stanislav Sykora, Jun 01 2014

STATUS

approved

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