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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).

%I A344122 #33 Apr 20 2023 09:52:49
%S A344122 4,12,40,36,112,120,108,352,336,400,360,324,832,1056,1120,1008,1200,
%T A344122 1080,972,2176,2496,3520,3136,3168,3360,4000,3024,3600,3240,2916,4864,
%U A344122 6528,8320,9856,7488,10560,9408,11200,9504,10080,12000,9072,10800,9720,8748,11776,14592,21760
%N 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).
%C A344122 Tree-able degree sequences are degree sequences that can be realized as trees [Stern].
%C A344122 The partitions of n-2 are given in nondecreasing order of length/lex.
%H A344122 Washington Bomfim, <a href="/A344122/b344122.txt">Table of n, a(n) for n = 2..9297</a> (Rows n = 2..27, flattened)
%H A344122 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 A344122 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A344122 Triangle T(n,k) begins:
%e A344122 n/k 1    2     3 ...
%e A344122 2     4;
%e A344122 3    12;
%e A344122 4    40,   36;
%e A344122 5   112,  120,  108;
%e A344122 6   352,  336,  400,  360,  324;
%e A344122 7   832, 1056, 1120, 1008, 1200, 1080,  972;
%e A344122 8  2176, 2496, 3520, 3136, 3168, 3360, 4000, 3024, 3600, 3240, 2916;
%e A344122   ...
%e A344122 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 A344122   [4, 1, 1, 1, 1]     o           7*2*2*2*2 = 112.
%e A344122                    ( ) ( )
%e A344122                    o o o o
%e A344122   [3, 2, 1, 1, 1]        o        5*3*2*2*2 = 120.
%e A344122                        / | \
%e A344122                    o--o  o  o
%e A344122   [2, 2, 2, 1, 1]  o--o--o--o--o  3*3*3*2*2 = 108.
%o A344122 (PARI) \\ Gives row n of triangle, n >= 2.
%o A344122 Row(n)={my(j=1, V=vector(numbpart(n-2))); forpart(P=n-2,
%o A344122 V[j] = prod(k = 1, #P, prime(P[k] + 1)); V[j] <<= (n-#P); j++ ); V };
%Y A344122 Cf. A000041, A000055, A003946 (last terms in rows), A215366, A265127 (first column).
%K A344122 nonn,look,tabf
%O A344122 2,1
%A A344122 _Washington Bomfim_, Jun 02 2021