A344122 Triangle T(n,k) read by rows in which n-th row gives all tree-able degree sequences S of n nodes encoded as Product_{k in S} prime(k); n >= 2, 1<= k <= A000041(n-2).

4, 12, 40, 36, 112, 120, 108, 352, 336, 400, 360, 324, 832, 1056, 1120, 1008, 1200, 1080, 972, 2176, 2496, 3520, 3136, 3168, 3360, 4000, 3024, 3600, 3240, 2916, 4864, 6528, 8320, 9856, 7488, 10560, 9408, 11200, 9504, 10080, 12000, 9072, 10800, 9720, 8748, 11776, 14592, 21760

Offset: 2

Author

Washington Bomfim, Jun 02 2021

Keywords

• nonn
• look
• tabf

Comments

Tree-able degree sequences are degree sequences that can be realized as trees [Stern].

The partitions of n-2 are given in nondecreasing order of length/lex.

Examples

			Triangle T(n,k) begins:
n/k 1    2     3 ...
2     4;
3    12;
4    40,   36;
5   112,  120,  108;
6   352,  336,  400,  360,  324;
7   832, 1056, 1120, 1008, 1200, 1080,  972;
8  2176, 2496, 3520, 3136, 3168, 3360, 4000, 3024, 3600, 3240, 2916;
  ...
Row 5 is 112, 120, 108 because prime(1) = 2, prime(2) = 3, prime(3) = 5, and prime(4) = 7. The tree-able degree sequences of 5 nodes, related tree realization and encode are given below.
  [4, 1, 1, 1, 1]     o           7*2*2*2*2 = 112.
                   ( ) ( )
                   o o o o
  [3, 2, 1, 1, 1]        o        5*3*2*2*2 = 120.
                       / | \
                   o--o  o  o
  [2, 2, 2, 1, 1]  o--o--o--o--o  3*3*3*2*2 = 108.
		

Links

• Washington Bomfim, Table of n, a(n) for n = 2..9297 (Rows n = 2..27, flattened)
• Samuel Stern, The Tree of Trees: on methods for finding all non-isomorphic tree-realizations of degree sequences, Honors Thesis, Wesleyan University, 2017.
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000041, A000055, A003946 (last terms in rows), A215366, A265127 (first column).

Programs

PARI

\\ Gives row n of triangle, n >= 2.
Row(n)={my(j=1, V=vector(numbpart(n-2))); forpart(P=n-2,
V[j] = prod(k = 1, #P, prime(P[k] + 1)); V[j] <<= (n-#P); j++ ); V };