A358107 -id:A358107 - cp's OEIS frontend

A055290 Triangle of trees with n nodes and k leaves, 2 <= k <= n.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 3, 4, 2, 1, 0, 1, 4, 8, 6, 3, 1, 0, 1, 5, 14, 14, 9, 3, 1, 0, 1, 7, 23, 32, 26, 12, 4, 1, 0, 1, 8, 36, 64, 66, 39, 16, 4, 1, 0, 1, 10, 54, 123, 158, 119, 60, 20, 5, 1, 0, 1, 12, 78, 219, 350, 325, 202, 83, 25, 5, 1, 0

Offset: 2

Christian G. Bower, May 09 2000

			Triangle begins:
  n=2:  1
  n=3:  1   0
  n=4:  1   1   0
  n=5:  1   1   1   0
  n=6:  1   2   2   1   0
  n=7:  1   3   4   2   1   0
  n=8:  1   4   8   6   3   1   0
  n=9:  1   5  14  14   9   3   1   0
  n=10: 1   7  23  32  26  12   4   1   0
  n=11: 1   8  36  64  66  39  16   4   1   0
  n=12: 1  10  54 123 158 119  60  20   5   1   0
  n=13: 1  12  78 219 350 325 202  83  25   5   1   0

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 80, Problem 3.9.

Andrew Howroyd, Table of n, a(n) for n = 2..1226 (rows 2..50)
Index entries for sequences related to trees

Row sums give A000055, row sums with weight k give A003228.
Columns 3 through 12: A001399(n-4), A055291, A055292, A055293, A055294, A055295, A055296, A055297, A055298, A055299.
Cf. A055300, A055301, A238416, A304222.
The labeled version is A055314.
Central column is A358107.
Left of central column is A359398.
Cf. A000088, A000272, A001349, A001433, A002494, A185650.

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
T(n)={my(u=[y]); for(n=2, n, u=concat([y], EulerMT(u))); my(r=x*Ser(u), v=Vec(r*(1-x+x*y) + (substvec(r,[x,y],[x^2,y^2]) - r^2)/2)); vector(n-1, k, Vecrev(v[1+k]/y^2, k))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

G.f.: A(x, y)=(1-x+x*y)*B(x, y)+(1/2)*(B(x^2, y^2)-B(x, y)^2), where B(x, y) is g.f. of A055277.

A358732 Number of labeled trees covering 2n nodes, half of which are leaves.

Original entry on oeis.org

0, 12, 720, 109200, 31752000, 15186346560, 10852244282880, 10851787634688000, 14481281691676800000, 24881574582258352358400, 53525038934303849706393600, 140958354488116955062668595200, 446153762528143389466306560000000, 1671353230826683972965623004979200000

Offset: 1

Gus Wiseman, Dec 01 2022

nonn

			The a(2) = 12 trees:
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,3},{3,4}}
  {{1,2},{1,4},{2,3}}
  {{1,2},{1,4},{3,4}}
  {{1,2},{2,3},{3,4}}
  {{1,2},{2,4},{3,4}}
  {{1,3},{1,4},{2,3}}
  {{1,3},{1,4},{2,4}}
  {{1,3},{2,3},{2,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4}}
  {{1,4},{2,3},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 1..100

A central column of A055314.
The unlabeled rooted version is A185650.
The unlabeled version is A358107.
A000272 counts trees, bisection A163395.
A001187 counts connected graphs.
A006129 counts covering graphs.
A014068 counts graphs with n vertices and n-1 edges.
Cf. A000055, A001349, A006125.

a[n_]:=StirlingS2[2*n-2, n]*(2*n)!/n!; Array[a,14] (* Stefano Spezia, Aug 02 2024 *)

a(n) = stirling(2*n-2, n, 2)*(2*n)!/n! \\ Andrew Howroyd, Dec 30 2022

a(n) = A055314(2*n, n) = Stirling2(2*n-2, n)*(2*n)!/n!. - Andrew Howroyd, Dec 30 2022

Terms a(6) and beyond from Andrew Howroyd, Dec 30 2022

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

Original entry on oeis.org

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

cp's OEIS Frontend

A055290 Triangle of trees with n nodes and k leaves, 2 <= k <= n.

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A358732 Number of labeled trees covering 2n nodes, half of which are leaves.

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

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