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A335362 Triangle T(n,d) read by rows: the number of mixed trees with n>=1 nodes and 0<=d arcs.

%I A335362 #22 Mar 23 2023 19:51:26
%S A335362 1,1,1,1,2,3,2,5,10,8,3,12,32,40,27,6,30,99,178,187,91,11,74,298,692,
%T A335362 1019,854,350,23,188,890,2538,4751,5692,4074,1376,47,478,2627,8886,
%U A335362 20260,31188,31856,19602,5743,106,1235,7734,30270,81170,152509,200413,177266,96035,24635
%N A335362 Triangle T(n,d) read by rows: the number of mixed trees with n>=1 nodes and 0<=d<n arcs.
%H A335362 Andrew Howroyd, <a href="/A335362/b335362.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows).
%H A335362 R. J. Mathar, <a href="/A335362/a335362.pdf">Mixed Trees A335362</a>
%H A335362 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%e A335362 The triangle starts
%e A335362 1;
%e A335362 1, 1;
%e A335362 1, 2, 3;
%e A335362 2, 5,10, 8;
%e A335362 3,12,32,40,27;
%e A335362 There are T(3,1)=2 mixed trees on 3 nodes with one directed edge (the edge can point towards the middle node or away from it).
%o A335362 (PARI) \\ Here R(n) is rooted mixed trees as g.f.
%o A335362 EulerMTS(p)={my(n=serprec(p,x)-1,vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i))}
%o A335362 R(n) = {my(p=x+O(x^2)); for(n=2, n, p=x*EulerMTS(2*y*p + p)); p}
%o A335362 T(n) = {my(p=R(n)); [Vecrev(p) | p<-Vec(p + (subst(subst(p + O(x*x^(n\2)), x, x^2), y, y^2) - (2*y+1)*p^2)/2)]}
%o A335362 { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Mar 23 2023
%Y A335362 Cf. A000055 (column d=0), A000238 (diagonal d=n-1), A000106 (column d=1), A006965 (row sums), A335601 (subdiagonal d=n-2).
%K A335362 nonn,tabl
%O A335362 1,5
%A A335362 _R. J. Mathar_, Jun 03 2020
%E A335362 Completed row n=9. - _R. J. Mathar_, Jun 11 2020
%E A335362 Terms a(46) and beyond from _Andrew Howroyd_, Mar 23 2023