A055302 -id:A055302 - cp's OEIS frontend

A231602 Triangular array read by rows: T(n,k) is the number of rooted labeled trees on n nodes that have exactly k nodes with outdegree = 1, n>=1, 0<=k<=n-1.

1, 0, 2, 3, 0, 6, 4, 36, 0, 24, 65, 80, 360, 0, 120, 306, 1950, 1200, 3600, 0, 720, 4207, 12852, 40950, 16800, 37800, 0, 5040, 38424, 235592, 359856, 764400, 235200, 423360, 0, 40320, 573057, 2766528, 8481312, 8636544, 13759200, 3386880, 5080320, 0, 362880

Offset: 1

Geoffrey Critzer, Nov 11 2013

T(n,k) is also the number of functions f:{1,2,...,n-1}->{1,2,...,n} that have exactly k elements whose preimage has cardinality = 1.
T(n,n-1) = n! = A000142(n).
Column k = 0 = A060356(n).
Row sums = n^(n-1) = A000169(n).
Refinement given by A248120. Sum coefficients of the partition polynomials with h_1 = (1') = t and all other h_n = (n') = 1 to obtain this entry. - Tom Copeland, Feb 01 2016

			1;
0, 2;
3, 0, 6;
4, 36, 0, 24;
65, 80, 360, 0, 120;
306, 1950, 1200, 3600, 0, 720;
4207, 12852, 40950, 16800, 37800, 0, 5040;
38424, 235592, 359856, 764400, 235200, 423360, 0, 40320;
....0..........0........
....|........./ \.......
....0........0...0......
.../ \.......|..........
..0   0......0..........
T(4,1) = 36.  Both of these graphs on 4 nodes have exactly 1 node that has outdegree = 1.  There are 12 + 24 = 36 labelings.

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

Cf. A055302, A206823.

Cf. A248120.

with(combinat): C:= binomial:
b:= proc(t, i, u) option remember; `if`(t=0, 1,
      `if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)
      *b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))
    end:
T:= (n, k)-> C(n, k)*C(n-1, k)*k! *b(n-1-k$2, n-k):
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Nov 12 2013

nn=8;Table[Table[Drop[Range[0,nn]!CoefficientList[Series[-ProductLog[x/(-1-x+x y)],{x,0,nn}],{x,y}],1][[r,c]],{c,1,r}],{r,1,nn}]//Grid

E.g.f. satisfies A(x,y) = y*x*A(x,y) + x*( exp(A(x,y)) - A(x,y) ).

A055313 Number of labeled rooted trees with n nodes and 12 leaves.

Original entry on oeis.org

13, 745290, 2153888100, 1850107896000, 813987254808000, 233983253596659840, 50146687652338368000, 8684014103185704192000, 1281139242561407305440000, 167044094432657294973120000

Offset: 13

Christian G. Bower, May 11 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 13..200
Index entries for sequences related to rooted trees

Column 12 of A055302.

[Factorial(n)*(n-12)^2*(n-11)^2*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n^8 - 708*n^7 + 24018*n^6 - 458136*n^5 + 5363513*n^4 - 39369364*n^3 + 176423596*n^2 - 439700816*n + 464486400) / 352423730384732160000: n in [13..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-12)^2*(n-11)^2*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n^8 - 708*n^7 + 24018*n^6 - 458136*n^5 + 5363513*n^4 - 39369364*n^3 + 176423596*n^2 - 439700816*n + 464486400) / 352423730384732160000,{n,13,25}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = n! * (n-12)^2*(n-11)^2*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n^8 - 708*n^7 + 24018*n^6 - 458136*n^5 + 5363513*n^4 - 39369364*n^3 + 176423596*n^2 - 439700816*n + 464486400) / 352423730384732160000. - Vaclav Kotesovec, Jul 25 2014

A055304 Number of labeled rooted trees with n nodes and 3 leaves.

Original entry on oeis.org

4, 140, 3000, 54600, 940800, 16087680, 279417600, 4989600000, 92207808000, 1769511744000, 35321737651200, 733823266176000, 15866448998400000, 356873052856320000, 8344427063132160000, 202660737281114112000

Offset: 4

Christian G. Bower, May 11 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 4..200
Index entries for sequences related to rooted trees

Column 3 of A055302.

[Factorial(n)*(n-3)*(n-2)*(n-1)*(3*n-8)/144 : n in [4..25]]; // Vincenzo Librandi, Jul 25 2014

seq(n!*(n-3)*(n-2)*(n-1)*(3*n-8)/144, n=4..100); # Robert Israel, Jul 25 2014

Table[n! * (n-3)*(n-2)*(n-1)*(3*n-8)/144,{n,4,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = (n!/3!)*Stirling2(n-1, n-3). - Vladeta Jovovic, Jan 28 2004
a(n) = n! * (n-3)*(n-2)*(n-1)*(3*n-8)/144. - Vaclav Kotesovec, Jul 25 2014
E.g.f.: x^4*(2*x+1)/(6*(1-x)^5). - Robert Israel, Jul 25 2014

A055305 Number of labeled rooted trees with n nodes and 4 leaves.

Original entry on oeis.org

5, 450, 18900, 588000, 15876000, 400075200, 9779616000, 237105792000, 5779453680000, 142845262560000, 3599700616512000, 92818726640640000, 2454539660052480000, 66664716210892800000, 1861170036255129600000

Offset: 5

Christian G. Bower, May 11 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 5..200
Index entries for sequences related to rooted trees

Column 4 of A055302.

[Factorial(n)*(n-4)^2*(n-3)^2*(n-2)*(n-1)/1152 : n in [5..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-4)^2*(n-3)^2*(n-2)*(n-1)/1152,{n,5,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = n! * (n-4)^2*(n-3)^2*(n-2)*(n-1)/1152. - Vaclav Kotesovec, Jul 25 2014

A055306 Number of labeled rooted trees with n nodes and 5 leaves.

Original entry on oeis.org

6, 1302, 101136, 5143824, 210198240, 7593173280, 255415628160, 8252203639680, 261173083691520, 8202954132172800, 257956727815987200, 8172268419809894400, 261973287647034163200, 8524055367713146060800

Offset: 6

Christian G. Bower, May 11 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 6..200
Index entries for sequences related to rooted trees

Column 5 of A055302.

[Factorial(n)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(15*n^3 - 195*n^2 + 830*n - 1152)/691200 : n in [6..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(15*n^3 - 195*n^2 + 830*n - 1152)/691200,{n,6,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = n! * (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(15*n^3 - 195*n^2 + 830*n - 1152)/691200. - Vaclav Kotesovec, Jul 25 2014

E.g.f: x^6*(1+22*x+58*x^2+24*x^3)/(120*(1-x)^9). - Robert Israel, Jul 25 2014

A055307 Number of labeled rooted trees with n nodes and 6 leaves.

Original entry on oeis.org

7, 3528, 486864, 39160800, 2357586000, 119409111360, 5426122141440, 230006844587520, 9326497051872000, 367969396354560000, 14295131088292454400, 551514022702420377600, 21263333439482226892800

Offset: 7

Christian G. Bower, May 11 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 7..200
Index entries for sequences related to rooted trees

Column 6 of A055302.

[Factorial(n)*(n-6)^2*(n-5)^2*(n-4)*(n-3)*(n-2)*(n-1)*(3*n^2-29*n+ 64)/8294400: n in [7..25]]; // Vincenzo Librandi, Jul 25 2014

seq(n! * (n-6)^2*(n-5)^2*(n-4)*(n-3)*(n-2)*(n-1)*(3*n^2 - 29*n + 64)/8294400, n=7..100); # Robert Israel, Jul 25 2014

Table[n! * (n-6)^2*(n-5)^2*(n-4)*(n-3)*(n-2)*(n-1)*(3*n^2 - 29*n + 64)/8294400,{n,7,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = n! * (n-6)^2*(n-5)^2*(n-4)*(n-3)*(n-2)*(n-1)*(3*n^2 - 29*n + 64)/8294400. - Vaclav Kotesovec, Jul 25 2014

E.g.f: x^7*(120*x^4+444*x^3+328*x^2+52*x+1)/(720*(1-x)^11). - Robert Israel, Jul 25 2014

A055308 Number of labeled rooted trees with n nodes and 7 leaves.

Original entry on oeis.org

8, 9144, 2178000, 270111600, 23449219200, 1635398519040, 98861289246720, 5425893817344000, 278673669001728000, 13673851998782976000, 650477329618365849600, 30322668859342072012800

Offset: 8

Christian G. Bower, May 11 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 8..200
Index entries for sequences related to rooted trees

Column 7 of A055302.

[Factorial(n)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(63*n^5- 1890*n^4+22365*n^3-130186*n^2+371672*n-414720)/14631321600: n in [8..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(63*n^5 - 1890*n^4 + 22365*n^3 - 130186*n^2 + 371672*n - 414720)/14631321600,{n,8,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = n! * (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(63*n^5 - 1890*n^4 + 22365*n^3 - 130186*n^2 + 371672*n - 414720)/14631321600. - Vaclav Kotesovec, Jul 25 2014

A055309 Number of labeled rooted trees with n nodes and 8 leaves.

Original entry on oeis.org

9, 22950, 9236700, 1731510000, 213034536000, 20154167953920, 1599866940672000, 112412172128256000, 7240640845838400000, 437993205197708160000, 25315591135036262860800

Offset: 9

Christian G. Bower, May 11 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 9..200
Index entries for sequences related to rooted trees

Column 8 of A055302.

[Factorial(n)*(n-8)^2*(n-7)^2*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n^4 - 234*n^3 + 2211*n^2 - 8938*n + 12960)/234101145600 : n in [9..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-8)^2*(n-7)^2*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n^4 - 234*n^3 + 2211*n^2 - 8938*n + 12960)/234101145600,{n,9,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = n! * (n-8)^2*(n-7)^2*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n^4 - 234*n^3 + 2211*n^2 - 8938*n + 12960)/234101145600. - Vaclav Kotesovec, Jul 25 2014

A055310 Number of labeled rooted trees with n nodes and 9 leaves.

Original entry on oeis.org

10, 56210, 37621320, 10493357160, 1803842280240, 228599313742800, 23567044281580800, 2099312576058297600, 168119161594140556800, 12440787291129176179200, 867641306901247726848000

Offset: 10

Christian G. Bower, May 11 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 10..200
Index entries for sequences related to rooted trees

Column 9 of A055302.

[Factorial(n)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(135*n^7 - 7245*n^6 + 164745*n^5 - 2054535*n^4 + 15148840*n^3 - 65892420*n^2 + 156108464*n - 154828800)/505658474496000: n in [10..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(135*n^7 - 7245*n^6 + 164745*n^5 - 2054535*n^4 + 15148840*n^3 - 65892420*n^2 + 156108464*n - 154828800)/505658474496000,{n,10,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = n! * (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(135*n^7 - 7245*n^6 + 164745*n^5 - 2054535*n^4 + 15148840*n^3 - 65892420*n^2 + 156108464*n - 154828800)/505658474496000. - Vaclav Kotesovec, Jul 25 2014

A055311 Number of labeled rooted trees with n nodes and 10 leaves.

Original entry on oeis.org

11, 135036, 148478616, 60841500720, 14441439612600, 2425616445732480, 321683306528263680, 36020400612748231680, 3559225274902019923200, 320000251320623922048000, 26762519467151924853196800

Offset: 11

Christian G. Bower, May 11 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 11..200
Index entries for sequences related to rooted trees

Column 10 of A055302.

[Factorial(n)*(n-10)^2*(n-9)^2*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(15*n^6 - 735*n^5 + 14715*n^4 - 153617*n^3 + 878862*n^2 - 2601592*n + 3096576) / 10113169489920000: n in [11..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-10)^2*(n-9)^2*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(15*n^6 - 735*n^5 + 14715*n^4 - 153617*n^3 + 878862*n^2 - 2601592*n + 3096576)/10113169489920000,{n,11,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = n! * (n-10)^2*(n-9)^2*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(15*n^6 - 735*n^5 + 14715*n^4 - 153617*n^3 + 878862*n^2 - 2601592*n + 3096576)/10113169489920000. - Vaclav Kotesovec, Jul 25 2014

cp's OEIS Frontend

A231602 Triangular array read by rows: T(n,k) is the number of rooted labeled trees on n nodes that have exactly k nodes with outdegree = 1, n>=1, 0<=k<=n-1.

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A055313 Number of labeled rooted trees with n nodes and 12 leaves.

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A055304 Number of labeled rooted trees with n nodes and 3 leaves.

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A055305 Number of labeled rooted trees with n nodes and 4 leaves.

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A055306 Number of labeled rooted trees with n nodes and 5 leaves.

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A055307 Number of labeled rooted trees with n nodes and 6 leaves.

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A055308 Number of labeled rooted trees with n nodes and 7 leaves.

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Magma

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A055309 Number of labeled rooted trees with n nodes and 8 leaves.

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