A303842 - OEIS

%I #17 May 21 2018 15:17:50

%S 1,1,1,1,1,2,1,2,3,3,1,2,6,6,6,1,3,9,15,16,11,1,3,13,26,43,37,23,1,4,

%T 17,46,88,116,96,47,1,4,23,68,169,273,329,239,106,1,5,28,103,287,585,

%U 869,918,622,235,1,5,35,141,467,1104,2031,2695,2609,1607,551

%N Triangle read by rows: T(s,n) (s>=1 and 2<=n<=s+1) = number of trees with n nodes and positive integer edge labels with label sum s.

%H Andrew Howroyd, <a href="/A303842/b303842.txt">Table of n, a(n) for n = 1..1275</a>

%H R. J. Mathar, <a href="http://viXra.org/abs/1805.0205">Labeled Trees with fixed node label sum</a>, vixra:1805.0205 (2018).

%e The triangle starts

%e 1;

%e 1   1;

%e 1   1   2;

%e 1   2   3    3;

%e 1   2   6    6    6;

%e 1   3   9    15   16    11;

%e 1   3   13   26   43    37     23;

%e 1   4   17   46   88    116    96    47;

%e 1   4   23   68   169   273    329   239  106;

%e 1   5   28   103  287   585    869   918  622    235;

%e 1   5   35   141  467   1104   2031  2695 2609   1607   551;

%e 1   6   42   195  711   1972   4211  6882 8399   ...    4235  1301;

%e 1   6   50   253  1051  3270   8108 15513 23152  ...    ... ;

%e 1   7   58   330  1489  5222  14552 32191 56291  ...    ... ;

%e 1   7   68   412  2063  7958  24846 62014 124958  ...    ... ;

%o (PARI)

%o EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}

%o b(n)={my(v=[1]); for(i=1, n, v=concat([1], v + EulerMT(y*v))); Ser(v)*y*(1-x)}

%o seq(n)={my(g=b(n)); Vec(g + (substvec(g, [x,y], [x^2,y^2]) - g^2)*x/(2*(1-x)) - y)}

%o {my(A=seq(15)); for(n=1, #A, print(Vecrev(A[n]/y^2)))} \\ _Andrew Howroyd_, May 20 2018

%Y Cf. A303841 (labeled nodes), A000055 (diagonal), A027852 (subdiagonal), A303833 (subdiagonal), A304914 (row sums).

%K nonn,tabl

%O 1,6

%A _R. J. Mathar_, May 01 2018