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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).
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%I #33 Apr 20 2023 09:52:49

%S 4,12,40,36,112,120,108,352,336,400,360,324,832,1056,1120,1008,1200,

%T 1080,972,2176,2496,3520,3136,3168,3360,4000,3024,3600,3240,2916,4864,

%U 6528,8320,9856,7488,10560,9408,11200,9504,10080,12000,9072,10800,9720,8748,11776,14592,21760

%N 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).

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

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

%H Washington Bomfim, <a href="/A344122/b344122.txt">Table of n, a(n) for n = 2..9297</a> (Rows n = 2..27, flattened)

%H Samuel Stern, <a href="https://digitalcollections.wesleyan.edu/object/ir%3A672">The Tree of Trees: on methods for finding all non-isomorphic tree-realizations of degree sequences</a>, Honors Thesis, Wesleyan University, 2017.

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%e Triangle T(n,k) begins:

%e n/k 1 2 3 ...

%e 2 4;

%e 3 12;

%e 4 40, 36;

%e 5 112, 120, 108;

%e 6 352, 336, 400, 360, 324;

%e 7 832, 1056, 1120, 1008, 1200, 1080, 972;

%e 8 2176, 2496, 3520, 3136, 3168, 3360, 4000, 3024, 3600, 3240, 2916;

%e ...

%e 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.

%e [4, 1, 1, 1, 1] o 7*2*2*2*2 = 112.

%e ( ) ( )

%e o o o o

%e [3, 2, 1, 1, 1] o 5*3*2*2*2 = 120.

%e / | \

%e o--o o o

%e [2, 2, 2, 1, 1] o--o--o--o--o 3*3*3*2*2 = 108.

%o (PARI) \\ Gives row n of triangle, n >= 2.

%o Row(n)={my(j=1, V=vector(numbpart(n-2))); forpart(P=n-2,

%o V[j] = prod(k = 1, #P, prime(P[k] + 1)); V[j] <<= (n-#P); j++ ); V };

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

%K nonn,look,tabf

%O 2,1

%A _Washington Bomfim_, Jun 02 2021