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==Various Topics==
==A055277 Triangle T(n,k) of number of rooted trees with n nodes and k leaves, 1 <= k <= n.==
* I recently got an [https://www.rechnen-ohne-strom.de/rechner-galerie/zahlenschieber/addimult/ ADDIMULT].
The triangle starts:
      A002 A550 A550 A550 A550 A550 A550 A550 A550 A550 A550 
        620  278  279  280  281  282  283  284  285  286  287
n/k 1    2    3    4    5    6    7    8    9  10  11  12
--+---------------------------------------------------------------
  1| 1
  2| 1    0
  3| 1    1    0
  4| 1    2    1    0
  5| 1    4    3    1    0
  6| 1    6    8    4    1    0
  7| 1    9  18  14    5    1    0
  8| 1  12  35  39  21    6    1    0
  9| 1  16  62  97  72  30    7    1    0
10| 1  20  103  212  214  120  40    8    1    0
11| 1  25  161  429  563  416  185  52    9    1    0
12| 1  30  241  804 1344 1268  732  270  65  10    1    0
13| 1  36  348 1427 2958 3499 2544 1203  378  80  11    1    0
For some columns, an o.g.f. is already given or conjectured:
A002620 (column 2): x^2 / ((1+x)*(1-x)^3)
A055278 (column 3): x^4*(x^3+x^2+1) / ((1-x^2)*(1-x^3)*(1-x)^3)
A055279 (column 4): x^5*(1+x+3*x^2+5*x^3+7*x^4+5*x^5+5*x^6+2*x^7+x^8) / ((1-x)^3*(1-x^2)^2*(1-x^3)*(1-x^4))
"Guessed" o.g.f.s (with Maple's <code>guessgf</code>) show denominators that are products of cyclotomic polynomials. The following factorizations of the denominators are obtained:
deo[2 ] :=                                                                                            (x^2-1)  *(x-1)^2;
deo[3 ] :=                                                                                  (x^3-1)  *(x^2-1)  *(x-1)^3;
deo[4 ] :=                                                                        (x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^3;
deo[5 ] :=                                                              (x^5-1)  *(x^4-1)            *(x^2-1)^2*(x-1)^5;
deo[6 ] :=                                                    (x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^3;
deo[7 ] :=                                            (x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^4;
deo[8 ] :=                                    (x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)  *(x^2-1)^3*(x-1)^5;
deo[9 ] :=                            (x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^3*(x^2-1)^3*(x-1)^4;
deo[10] :=                    (x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^4*(x-1)^3;
deo[11] :=          (x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^2*(x^2-1)^4*(x-1)^5;
deo[12] :=  (x^12-1)*(x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)^2*(x^5-1)^2*(x^4-1)^3*(x^3-1)^3*(x^2-1)^3*(x-1)^4;
Some pattern can be seen, but the higher exponents are not systematic. The problem is that some factors may have been cancelled out in the o.g.f.s polynomial fraction.
However, by multiplying both the nominator and the denominator polynomials with such factors, the following form of the denominators can be obtained:
den[2 ] :=                                                                                            (x^2-1)  *(x-1)^2 ;
den[3 ] :=                                                                                  (x^3-1)  *(x^2-1)  *(x-1)^3 ;
den[4 ] :=                                                                        (x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^'''<span style="color:red">4</span>''' ;
den[5 ] :=                                                              (x^5-1)  *(x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^5 ;
den[6 ] :=                                                    (x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^'''<span style="color:red">6</span>''' ;
den[7 ] :=                                            (x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^'''<span style="color:red">7</span>''' ;
den[8 ] :=                                    (x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^'''<span style="color:red">2</span>'''*(x^2-1)^'''<span style="color:red">4</span>'''*(x-1)^'''<span style="color:red">8</span>''' ;
den[9 ] :=                            (x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^3*(x^2-1)^'''<span style="color:red">4</span>'''*(x-1)^'''<span style="color:red">9</span>''' ;
den[10] :=                    (x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^'''<span style="color:red">5</span>'''*(x-1)^'''<span style="color:red">10</span>''';
den[11] :=          (x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^'''<span style="color:red">3</span>'''*(x^2-1)^'''<span style="color:red">5</span>'''*(x-1)^'''<span style="color:red">11</span>''';
den[12] :=  (x^12-1)*(x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)^2*(x^5-1)^2*(x^4-1)^3*(x^3-1)^'''<span style="color:red">4</span>'''*(x^2-1)^'''<span style="color:red">6</span>'''*(x-1)^'''<span style="color:red">12</span>''';
Here, the exponents increase in groups of 1 for <code>(x-1)</code>, in groups of 2 for <code>(x^2-1)</code>, in groups of 3 for <code>(x^3-1)</code>, etc.
Please note that this widening of the g.f.'s denominator polynomials must also be done for the numerator polynomials, but that the initial terms of the sequence remain unaffected.
The advantage of this deliberate increase of degree in both polynomials is the simple and easy formula for the exponents of <code>(x^m-1)</code> in the denominators.
 
These exponents turn out to be the reversed rows of A010766. The latter has a very simple formula <code>T(n,k) = floor(n/k)</code> and starts:
  n/k  1  2  3  4  5  6  7  8  9 10 11 12
  ---+------------------------------------
    1 |  1
    2 |  2  1
    3 |  3  1  1
    4 |  4  2  1  1
    5 |  5  2  1  1  1
    6 |  6  3  2  1  1  1
    7 |  7  3  2  1  1  1  1
    8 |  8  4  2  2  1  1  1  1
    9 |  9  4  3  2  1  1  1  1  1
  10 | 10  5  3  2  2  1  1  1  1  1
  11 | 11  5  3  2  2  1  1  1  1  1  1
  12 | 12  6  4  3  2  2  1  1  1  1  1  1
 
In order to support this hypothesis, the following steps were performed:
# Determine the denomiator polynomials <code>den</code> above, and take their coefficients as the signature for a linear recurrence with constant coefficients.
# Compute a sufficient number of terms (150 - 270) for the column sequences A055278-A055287 with Michale Somos' PARI program in A055277. This took about 1 day.
# Take <code>degree(den[n])</code> initial terms from step 2, run the linear recurrence and verify it for all terms.
 
In fact, the initial terms could have been been prefixed by <code>n</code> zeros, but in the OEIS sequences the leading zeros are omitted.

Latest revision as of 18:10, 6 March 2022

A055277 Triangle T(n,k) of number of rooted trees with n nodes and k leaves, 1 <= k <= n.

The triangle starts: 

      A002 A550 A550 A550 A550 A550 A550 A550 A550 A550 A550  
       620  278  279  280  281  282  283  284  285  286  287
n/k 1    2    3    4    5    6    7    8    9   10   11   12
--+---------------------------------------------------------------
 1| 1
 2| 1    0
 3| 1    1    0
 4| 1    2    1    0
 5| 1    4    3    1    0
 6| 1    6    8    4    1    0
 7| 1    9   18   14    5    1    0
 8| 1   12   35   39   21    6    1    0
 9| 1   16   62   97   72   30    7    1    0
10| 1   20  103  212  214  120   40    8    1    0
11| 1   25  161  429  563  416  185   52    9    1    0
12| 1   30  241  804 1344 1268  732  270   65   10    1    0
13| 1   36  348 1427 2958 3499 2544 1203  378   80   11    1    0

For some columns, an o.g.f. is already given or conjectured: 

A002620 (column 2): x^2 / ((1+x)*(1-x)^3)
A055278 (column 3): x^4*(x^3+x^2+1) / ((1-x^2)*(1-x^3)*(1-x)^3)
A055279 (column 4): x^5*(1+x+3*x^2+5*x^3+7*x^4+5*x^5+5*x^6+2*x^7+x^8) / ((1-x)^3*(1-x^2)^2*(1-x^3)*(1-x^4))

"Guessed" o.g.f.s (with Maple's guessgf) show denominators that are products of cyclotomic polynomials. The following factorizations of the denominators are obtained: 

deo[2 ] :=                                                                                             (x^2-1)  *(x-1)^2;
deo[3 ] :=                                                                                   (x^3-1)  *(x^2-1)  *(x-1)^3;
deo[4 ] :=                                                                         (x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^3;
deo[5 ] :=                                                               (x^5-1)  *(x^4-1)            *(x^2-1)^2*(x-1)^5;
deo[6 ] :=                                                     (x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^3;
deo[7 ] :=                                             (x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^4;
deo[8 ] :=                                     (x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)  *(x^2-1)^3*(x-1)^5;
deo[9 ] :=                             (x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^3*(x^2-1)^3*(x-1)^4;
deo[10] :=                    (x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^4*(x-1)^3;
deo[11] :=           (x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^2*(x^2-1)^4*(x-1)^5;
deo[12] :=  (x^12-1)*(x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)^2*(x^5-1)^2*(x^4-1)^3*(x^3-1)^3*(x^2-1)^3*(x-1)^4;

Some pattern can be seen, but the higher exponents are not systematic. The problem is that some factors may have been cancelled out in the o.g.f.s polynomial fraction. However, by multiplying both the nominator and the denominator polynomials with such factors, the following form of the denominators can be obtained: 

den[2 ] :=                                                                                             (x^2-1)  *(x-1)^2 ;
den[3 ] :=                                                                                   (x^3-1)  *(x^2-1)  *(x-1)^3 ;
den[4 ] :=                                                                         (x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^4 ;
den[5 ] :=                                                               (x^5-1)  *(x^4-1)  *(x^3-1)  *(x^2-1)^2*(x-1)^5 ;
den[6 ] :=                                                     (x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^6 ;
den[7 ] :=                                             (x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)  *(x^3-1)^2*(x^2-1)^3*(x-1)^7 ;
den[8 ] :=                                     (x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^2*(x^2-1)^4*(x-1)^8 ;
den[9 ] :=                             (x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)  *(x^4-1)^2*(x^3-1)^3*(x^2-1)^4*(x-1)^9 ;
den[10] :=                    (x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^5*(x-1)^10;
den[11] :=           (x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)  *(x^5-1)^2*(x^4-1)^2*(x^3-1)^3*(x^2-1)^5*(x-1)^11;
den[12] :=  (x^12-1)*(x^11-1)*(x^10-1)*(x^9-1)*(x^8-1)*(x^7-1)*(x^6-1)^2*(x^5-1)^2*(x^4-1)^3*(x^3-1)^4*(x^2-1)^6*(x-1)^12;

Here, the exponents increase in groups of 1 for (x-1), in groups of 2 for (x^2-1), in groups of 3 for (x^3-1), etc. Please note that this widening of the g.f.'s denominator polynomials must also be done for the numerator polynomials, but that the initial terms of the sequence remain unaffected. The advantage of this deliberate increase of degree in both polynomials is the simple and easy formula for the exponents of (x^m-1) in the denominators.

These exponents turn out to be the reversed rows of A010766. The latter has a very simple formula T(n,k) = floor(n/k) and starts: 

  n/k   1  2  3  4  5  6  7  8  9 10 11 12
  ---+------------------------------------
   1 |  1
   2 |  2  1
   3 |  3  1  1
   4 |  4  2  1  1
   5 |  5  2  1  1  1
   6 |  6  3  2  1  1  1
   7 |  7  3  2  1  1  1  1
   8 |  8  4  2  2  1  1  1  1
   9 |  9  4  3  2  1  1  1  1  1
  10 | 10  5  3  2  2  1  1  1  1  1
  11 | 11  5  3  2  2  1  1  1  1  1  1
  12 | 12  6  4  3  2  2  1  1  1  1  1  1

In order to support this hypothesis, the following steps were performed:

  1. Determine the denomiator polynomials den above, and take their coefficients as the signature for a linear recurrence with constant coefficients.
  2. Compute a sufficient number of terms (150 - 270) for the column sequences A055278-A055287 with Michale Somos' PARI program in A055277. This took about 1 day.
  3. Take degree(den[n]) initial terms from step 2, run the linear recurrence and verify it for all terms.

In fact, the initial terms could have been been prefixed by n zeros, but in the OEIS sequences the leading zeros are omitted.

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