A055314 -id:A055314 - cp's OEIS frontend

A359398 Number of unlabeled trees covering 2n nodes, half of which are leaves.

0, 1, 2, 8, 32, 158, 833, 4755, 28389, 176542, 1131055, 7432876, 49873477, 340658595, 2362652648, 16605707901, 118082160358, 848399575321, 6152038125538, 44981009272740, 331344933928536, 2457372361637286, 18337490246234464, 137612955519565773, 1038076541372187991

Offset: 1

Gus Wiseman, Jan 01 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100
Gus Wiseman, The a(4) = 8 trees covering 8 nodes, half of which are leaves.

Left of central column of A055290.
The labeled version is the left of central column of A055314.
The rooted version is A185650.
For n+1 leaves we have A358107.
The labeled version is A358732.
A000272 counts trees, bisection A163395, unlabeled A000055.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts graphs with n vertices and n-1 edges, unlabeled A001433.

a(n) = A055290(2*n, n). - Andrew Howroyd, Jan 01 2023

Terms a(12) and beyond from Andrew Howroyd, Jan 01 2023

A055315 Number of labeled trees with n nodes and 3 leaves.

Original entry on oeis.org

4, 60, 720, 8400, 100800, 1270080, 16934400, 239500800, 3592512000, 57081024000, 958961203200, 16999766784000, 317328979968000, 6224529991680000, 128047474114560000, 2757288942600192000, 62039001208504320000, 1456091851893719040000, 35593356379624243200000

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 trees

Column 3 of A055314.

[Factorial(n)*(n-3)*(n-2)/12: n in [4..25]]; // Vincenzo Librandi, Jul 25 2014

a:=n->sum((n-j)*n!/3!, j=3..n): seq(a(n), n=4..19); # Zerinvary Lajos, Apr 29 2007

Table[n!*(n-3)*(n-2)/12,{n,4,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

for(n=4,30, print1(n!*(n-3)*(n-2)/12, ", ")) \\ G. C. Greubel, Feb 07 2018

a(n) = (n!/3!)*stirling2(n-2, n-3). - Vladeta Jovovic, Jan 28 2004
a(n) = n! * (n-3)*(n-2)/12. - Vaclav Kotesovec, Jul 25 2014
E.g.f.: x*(x/(1-x))^3/3! - Geoffrey Critzer, Sep 19 2017

A055324 Number of labeled trees with n nodes and 12 leaves.

Original entry on oeis.org

13, 372554, 714236250, 453911421600, 156507084115200, 36555247168352640, 6528715119143118720, 960135043767367104000, 122086105154945279712000, 13885903109630633425344000, 1447862009053077400092710400, 140958354488116955062668595200

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 trees

Column 12 of A055314.

[Factorial(n)*(n-12)*(n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(99*n^9 - 9207*n^8 + 377586*n^7 - 8955870*n^6 + 135276603*n^5 - 1348112183*n^4 + 8853485696*n^3 - 36897359092*n^2 + 88399944688*n - 92577669120) / 176211865192366080000: n in [13..25]]; // Vincenzo Librandi, Jul 25 2014

[(Factorial(n)/Factorial(12))*StirlingSecond(n-2, n-12): n in [13..30]]; // G. C. Greubel, Feb 07 2018

Table[n! * (n-12)*(n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(99*n^9 - 9207*n^8 + 377586*n^7 - 8955870*n^6 + 135276603*n^5 - 1348112183*n^4 + 8853485696*n^3 - 36897359092*n^2 + 88399944688*n - 92577669120) / 176211865192366080000,{n,13,25}] (* Vaclav Kotesovec, Jul 25 2014 *)
Table[(n!/12!)*StirlingS2[n-2, n-12], {n,13, 30}] (* G. C. Greubel, Feb 07 2018 *)

for(n=13, 30, print1((n!/12!)*stirling(n-2, n-12, 2), ", ")) \\ G. C. Greubel, Feb 07 2018

a(n) = (n!/12!)*Stirling2(n-2, n-12). - Vladeta Jovovic, Jan 28 2004

a(n) = n! * (n-12)*(n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(99*n^9 - 9207*n^8 + 377586*n^7 - 8955870*n^6 + 135276603*n^5 - 1348112183*n^4 + 8853485696*n^3 - 36897359092*n^2 + 88399944688*n - 92577669120) / 176211865192366080000. - Vaclav Kotesovec, Jul 25 2014

Missing a(24) inserted by Andrew Howroyd, Feb 23 2018

A055316 Number of labeled trees with n nodes and 4 leaves.

Original entry on oeis.org

5, 210, 5250, 109200, 2116800, 40219200, 768398400, 14968800000, 299675376000, 6193291104000, 132456516192000, 2935293064704000, 67432408243200000, 1605928737853440000, 39636028549877760000, 1013303686405570560000, 26816358272375992320000, 734112975329750016000000

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 trees

Column 4 of A055314.

[Factorial(n)*(n-4)*(n-3)*(n-2)*(3*n-11)/576: n in [5..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-4)*(n-3)*(n-2)*(3*n-11)/576,{n,5,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = (n!/4!)*Stirling2(n-2, n-4). - Vladeta Jovovic, Jan 28 2004

a(n) = n! * (n-4)*(n-3)*(n-2)*(3*n-11)/576. - Vaclav Kotesovec, Jul 25 2014

A055317 Number of labeled trees with n nodes and 5 leaves.

Original entry on oeis.org

6, 630, 30240, 1058400, 31752000, 880165440, 23471078400, 616475059200, 16182470304000, 428535787680000, 11519041972838400, 315583670578176000, 8836342776188928000, 253325921601392640000, 7444680145020518400000

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 trees

Column 5 of A055314.

[Factorial(n)*(n-5)^2*(n-4)^2*(n-3)*(n-2)/5760: n in [6..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-5)^2*(n-4)^2*(n-3)*(n-2)/5760,{n,6,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

(n!/5!)*Stirling2(n-2, n-5). - Vladeta Jovovic, Jan 28 2004

a(n) = n! * (n-5)^2*(n-4)^2*(n-3)*(n-2)/5760. - Vaclav Kotesovec, Jul 25 2014

A055318 Number of labeled trees with n nodes and 6 leaves.

Original entry on oeis.org

7, 1736, 151704, 8573040, 385363440, 15186346560, 553400527680, 19255141825920, 652932709228800, 21874544352460800, 730877395478630400, 24516805259429683200, 829582077548941516800, 28413517892377153536000

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 trees

Column 6 of A055314.

[Factorial(n)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(15*n^3 - 240*n^2 + 1265*n - 2192)/4147200: n in [7..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(15*n^3 - 240*n^2 + 1265*n - 2192)/4147200,{n,7,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

(n!/6!)*Stirling2(n-2, n-6). - Vladeta Jovovic, Jan 28 2004

a(n) = n! * (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(15*n^3 - 240*n^2 + 1265*n - 2192)/4147200. - Vaclav Kotesovec, Jul 25 2014

A055319 Number of labeled trees with n nodes and 7 leaves.

Original entry on oeis.org

8, 4536, 695520, 61538400, 4041576000, 221759778240, 10852244282880, 492871809830400, 21317707547136000, 893639962575360000, 36758908512752025600, 1496966633049426739200, 60752381255663505408000, 2469167757848774062080000, 100876656745052194406400000

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 trees

Column 7 of A055314.

[Factorial(n)*(n-7)^2*(n-6)^2*(n-5)*(n-4)*(n-3)*(n-2)*(3*n^2 - 35*n + 96)/58060800: n in [8..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-7)^2*(n-6)^2*(n-5)*(n-4)*(n-3)*(n-2)*(3*n^2 - 35*n + 96)/58060800,{n,8,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = (n!/7!)*Stirling2(n-2, n-7). - Vladeta Jovovic, Jan 28 2004

a(n) = n! * (n-7)^2*(n-6)^2*(n-5)*(n-4)*(n-3)*(n-2)*(3*n^2 - 35*n + 96)/58060800. - Vaclav Kotesovec, Jul 25 2014

A055320 Number of labeled trees with n nodes and 8 leaves.

Original entry on oeis.org

9, 11430, 2994750, 405167400, 38104981200, 2861947408320, 185364917337600, 10851787634688000, 592181546628672000, 30766166997261696000, 1544883657843618892800, 75806672148355180032000

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 trees

Column 8 of A055314.

[Factorial(n)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(63*n^5 - 2205*n^4 + 30555*n^3 - 209251*n^2 + 707014*n - 940896)/117050572800: n in [9..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(63*n^5 - 2205*n^4 + 30555*n^3 - 209251*n^2 + 707014*n - 940896)/117050572800,{n,9,20}] (* Vaclav Kotesovec, Jul 25 2014 *)

(n!/8!)*Stirling2(n-2, n-8). - Vladeta Jovovic, Jan 28 2004

a(n) = n! * (n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(63*n^5 - 2205*n^4 + 30555*n^3 - 209251*n^2 + 707014*n - 940896)/117050572800. - Vaclav Kotesovec, Jul 25 2014

A055321 Number of labeled trees with n nodes and 9 leaves.

Original entry on oeis.org

10, 28050, 12315600, 2501070000, 331387056000, 33590279923200, 2844207894528000, 212334102908928000, 14481281691676800000, 924652322084050560000, 56256869188969473024000, 3303981073122303974400000, 189156797595688810567680000, 10636600593905858347776000000

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 trees

Column 9 of A055314.

[Factorial(n)*(n-9)^2*(n-8)^2*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(9*n^4 - 270*n^3 + 2967*n^2 - 14098*n + 24352)/2106910310400: n in [10..25]]; // Vincenzo Librandi, Jul 25 2014

a:= n-> (n!/9!)*Stirling2(n-2, n-9):
seq(a(n), n=10..25);  # Alois P. Heinz, Mar 06 2012

Table[n! * (n-9)^2*(n-8)^2*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(9*n^4 - 270*n^3 + 2967*n^2 - 14098*n + 24352)/2106910310400,{n,10,25}] (* Vaclav Kotesovec, Jul 25 2014 *)

A055321(n) := block(
        A055314(n,9)
)$
for n : 10 thru 25 do
        print(A055321(n)," ") ; /* R. J. Mathar, Mar 06 2012 */

A055321(n)={binomial(n,9)*sum(i=0,n-=9,(-1)^i*binomial(n,i)*i^(n+7))*(-1)^n} /* or: Stirling2(n-2, n-9)*n!/9!, cf. A008277 */ /* M. F. Hasler, Mar 06 2012 */

a(n) = (n!/9!)*Stirling2(n-2, n-9). - Vladeta Jovovic, Jan 28 2004

a(n) = n! * (n-9)^2*(n-8)^2*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(9*n^4 - 270*n^3 + 2967*n^2 - 14098*n + 24352)/2106910310400. - Vaclav Kotesovec, Jul 25 2014

A055322 Number of labeled trees with n nodes and 10 leaves.

Original entry on oeis.org

11, 67452, 48907716, 14690700024, 2705763420360, 365758901988480, 40063975278687360, 3778762636904935680, 319426407028867057920, 24881574582258352358400, 1822046744492620226380800

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 trees

Column 10 of A055314.

[Factorial(n)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(135*n^7 - 8190*n^6 + 211050*n^5 - 2991660*n^4 + 25164055*n^3 - 125425110*n^2 + 342426104*n - 394205184) / 5056584744960000: n in [11..25]]; // Vincenzo Librandi, Jul 25 2014

Table[n! * (n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(135*n^7 - 8190*n^6 + 211050*n^5 - 2991660*n^4 + 25164055*n^3 - 125425110*n^2 + 342426104*n - 394205184) / 5056584744960000,{n,11,25}] (* Vaclav Kotesovec, Jul 25 2014 *)

a(n) = (n!/10!)*Stirling2(n-2, n-10). - Vladeta Jovovic, Jan 28 2004

a(n) = n! * (n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(135*n^7 - 8190*n^6 + 211050*n^5 - 2991660*n^4 + 25164055*n^3 - 125425110*n^2 + 342426104*n - 394205184) / 5056584744960000. - Vaclav Kotesovec, Jul 25 2014

cp's OEIS Frontend

A359398 Number of unlabeled trees covering 2n nodes, half of which are leaves.

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A055315 Number of labeled trees with n nodes and 3 leaves.

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Magma

Maple

Mathematica

PARI

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A055324 Number of labeled trees with n nodes and 12 leaves.

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Magma

Magma

Mathematica

PARI

Formula

Extensions

A055316 Number of labeled trees with n nodes and 4 leaves.

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Magma

Mathematica

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A055317 Number of labeled trees with n nodes and 5 leaves.

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Magma

Mathematica

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A055318 Number of labeled trees with n nodes and 6 leaves.

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Magma

Mathematica

Formula

A055319 Number of labeled trees with n nodes and 7 leaves.

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Programs

Magma

Mathematica

Formula

A055320 Number of labeled trees with n nodes and 8 leaves.

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