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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).
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
2,1
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.
EXAMPLE
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.
PROG
(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 };
CROSSREFS
Cf. A000041, A000055, A003946 (last terms in rows), A215366, A265127 (first column).
Sequence in context: A149331 A009532 A248325 * A328533 A265127 A056274
KEYWORD
nonn,look,tabf
AUTHOR
Washington Bomfim, Jun 02 2021
STATUS
approved