A000106 -id:A000106 - cp's OEIS frontend

A370773 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes rooted at a hedron with a designated front.

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 2, 1, 0, 1, 12, 7, 2, 1, 0, 1, 30, 26, 7, 2, 1, 0, 1, 74, 110, 30, 7, 2, 1, 0, 1, 188, 481, 159, 30, 7, 2, 1, 0, 1, 478, 2209, 940, 168, 30, 7, 2, 1, 0, 1, 1235, 10379, 6093, 1104, 168, 30, 7, 2, 1, 0, 1, 3214, 49868, 41367, 8428, 1124, 168, 30, 7, 2, 1, 0

Offset: 0

Andrew Howroyd, Mar 01 2024

			Triangle begins:
  0;
  1,    0;
  1,    1,     0;
  1,    2,     1,    0;
  1,    5,     2,    1,    0;
  1,   12,     7,    2,    1,   0;
  1,   30,    26,    7,    2,   1,  0;
  1,   74,   110,   30,    7,   2,  1, 0;
  1,  188,   481,  159,   30,   7,  2, 1, 0;
  1,  478,  2209,  940,  168,  30,  7, 2, 1, 0;
  1, 1235, 10379, 6093, 1104, 168, 30, 7, 2, 1, 0;
  ...

Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45.

Columns k=1 is A000106.

Cf. A370770 (unrooted), A370771, A370772.

A304068 Number of trees on n vertices rooted at an oriented non-edge.

Original entry on oeis.org

0, 0, 1, 4, 14, 45, 140, 424, 1269, 3760, 11080, 32517, 95190, 278154, 811887, 2367973, 6903453, 20120905, 58639016, 170894228, 498084608, 1451899005, 4232957241, 12343454790, 36001675800, 105028397290, 306472665459, 894497511566

Offset: 1

Brendan McKay, May 05 2018

nonn

			a(4)=4 is based on the same examples as in A304067, but the oriented edge that spans a leaf with the node at distance 2 in the linear graph may have 2 orientations, so a(4) is one larger than A304067(4).

Cf. A000055 (not rooted), A000106 (rooted at oriented edge)

A000106(n) + a(n) = A000243(n).

cp's OEIS Frontend

A370773 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes rooted at a hedron with a designated front.

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