cp's OEIS Frontend

A345970 Irregular triangle T(n,k) read by rows in which n-th row lists in colex order all series-reduced tree degree sequences D of n nodes encoded as t = Product_{d in D} prime(d); n >= 4, 1 <= k <= A002865(n-2).

%I A345970 #100 Sep 29 2021 13:05:57
%S A345970 40,112,352,400,832,1120,2176,3520,3136,4000,4864,8320,9856,11200,
%T A345970 11776,21760,23296,30976,35200,31360,40000,29696,48640,60928,73216,
%U A345970 83200,98560,87808,112000,63488,117760,136192,191488,173056,217600,232960,309760,275968,352000,313600,400000
%N A345970 Irregular triangle T(n,k) read by rows in which n-th row lists in colex order all series-reduced tree degree sequences D of n nodes encoded as t = Product_{d in D} prime(d); n >= 4, 1 <= k <= A002865(n-2).
%C A345970 Tree degree sequences of n nodes are in one-to-one correspondence with the partitions of n-2, as for instance set out in Myerson's collection of problems [Myerson]. For series-reduced trees, these partitions have no part 1.
%C A345970 Given a term t, the respective degree sequence D is determined by Decode(t). See second (PARI) entry.
%C A345970 A250308(n) = Sum_{k= 1 .. A002865(2*n-2) } ( A345971(2*n,k) * odd( Decode( T(2*n,k) ) ), where odd(D) is 1 if all d in D are odd, and 0 otherwize.
%H A345970 Gerry Myerson, <a href="https://www.math.colostate.edu/~achter/wntc/problems/problems2004.pdf">Problems 2004</a>
%H A345970 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A345970 Triangle begins:
%e A345970   n \ k|  1    2 ...           n \ k| 1                2            ...
%e A345970   -----+-------------          -----+-----------------------------------
%e A345970   4    |   40;                 4    |       [3,1,1,1];
%e A345970   5    |  112;                 5    |     [4,1,1,1,1];
%e A345970   6    |  352,  400;    <=>    6    |   [5,1,1,1,1,1],   [3,3,1,1,1,1];
%e A345970   7    |  832, 1120;           7    | [6,1,1,1,1,1,1], [4,3,1,1,1,1,1];
%e A345970   ...                          ...
%e A345970 Row n = 7 follows from table
%e A345970                                                                          .
%e A345970   +---------------------+------------------+---------------------------+
%e A345970   | Partitions of n-2 = |                  |                           |
%e A345970   | 5 without parts 1   | Degree sequences |       Encoding            |
%e A345970   +---------------------+------------------+---------------------------+
%e A345970   |                 [5] |    6,1,1,1,1,1,1 |            prime(6) * 2^6 |
%e A345970   |              [2, 3] |    4,3,1,1,1,1,1 | prime(4) * prime(3) * 2^5 |
%e A345970   +---------------------+------------------+---------------------------+
%o A345970 (PARI) Row(n) = {my(j=0, V = vector(numbpart(n-2) - numbpart(n-3)));
%o A345970 forpart(P=n-2, V[j++] = prod(k=1,#P, prime(P[k]+1)) << (n-#P),[2, n-2]); V};
%o A345970 (PARI) Decode(t) = {my(V = [], i = 1, p); while(t > 1, p = prime(i); while(t % p == 0, t /= p; V = concat(V, Vec(i)) ); i++); vecsort(V, (x,y)->y-x) };
%Y A345970 Cf. A002865 (row widths), A265127 (column k=1), A345971 (number of trees by degree sequence), A344122 (free tree degree sequences), A250308.
%K A345970 nonn,look,tabf,easy
%O A345970 4,1
%A A345970 _Washington Bomfim_, Jul 01 2021