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A055327 Triangle of rooted identity trees with n nodes and k leaves.

%I A055327 #26 Aug 28 2018 23:38:33
%S A055327 1,1,1,1,1,1,2,1,4,1,1,6,5,1,9,13,2,1,12,28,11,1,16,53,40,3,1,20,91,
%T A055327 109,26,1,25,146,254,116,6,1,30,223,524,387,61,1,36,326,998,1068,329,
%U A055327 12,1,42,461,1774,2587,1289,145,1,49,634,2995,5678,4133,911,25,1,56
%N A055327 Triangle of rooted identity trees with n nodes and k leaves.
%C A055327 Row lengths are 1,1,1,2,2,3,3,4,4,5,5,6,6,...
%H A055327 Andrew Howroyd, <a href="/A055327/b055327.txt">Table of n, a(n) for n = 1..1226</a> (first 70 rows)
%H A055327 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A055327 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A055327 G.f. satisfies A(x,y) = x*y + x*WEIGH(A(x,y)) - x. Shifts up under WEIGH transform.
%e A055327 Triangle begins:
%e A055327 1;
%e A055327 1;
%e A055327 1;
%e A055327 1,  1;
%e A055327 1,  2;
%e A055327 1,  4,  1;
%e A055327 1,  6,  5;
%e A055327 1,  9, 13,  2;
%e A055327 1, 12, 28, 11;
%e A055327 1, 16, 53, 40, 3;
%e A055327 ...
%e A055327 From _Joerg Arndt_, Aug 18 2014: (Start)
%e A055327 The identity trees with n=6 nodes, as (preorder-) level sequences, together with their number of leaves, and an ASCII rendering, are:
%e A055327 :
%e A055327 :     1:  [ 0 1 2 3 4 5 ]   1
%e A055327 :  O--o--o--o--o--o
%e A055327 :
%e A055327 :     2:  [ 0 1 2 3 4 3 ]   2
%e A055327 :  O--o--o--o--o
%e A055327 :        .--o
%e A055327 :
%e A055327 :     3:  [ 0 1 2 3 4 2 ]   2
%e A055327 :  O--o--o--o--o
%e A055327 :     .--o
%e A055327 :
%e A055327 :     4:  [ 0 1 2 3 4 1 ]   2
%e A055327 :  O--o--o--o--o
%e A055327 :  .--o
%e A055327 :
%e A055327 :     5:  [ 0 1 2 3 2 1 ]   3
%e A055327 :  O--o--o--o
%e A055327 :     .--o
%e A055327 :  .--o
%e A055327 :
%e A055327 :     6:  [ 0 1 2 3 1 2 ]   2
%e A055327 :  O--o--o--o
%e A055327 :  .--o--o
%e A055327 :
%e A055327 This gives [1, 4, 1], row n=6 of the triangle.
%e A055327 (End)
%o A055327 (PARI)
%o A055327 WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
%o A055327 A(n)={my(v=[y]); for(n=2, n, v=concat([y], WeighMT(v))); apply(p->Vecrev(p/y), v)}
%o A055327 { my(T=A(15)); for(n=1, #T, print(T[n])) } \\ _Andrew Howroyd_, Aug 28 2018
%Y A055327 Row sums give A004111.
%Y A055327 Columns 2 to 8: A002620(n-2), A055328, A055329, A055330, A055331, A055332, A055333.
%Y A055327 A regular version is A301342.
%Y A055327 Cf. A055334.
%K A055327 nonn,tabf,eigen
%O A055327 1,7
%A A055327 _Christian G. Bower_, May 12 2000