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A055327
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Triangle of rooted identity trees with n nodes and k leaves.
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12
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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, 109, 26, 1, 25, 146, 254, 116, 6, 1, 30, 223, 524, 387, 61, 1, 36, 326, 998, 1068, 329, 12, 1, 42, 461, 1774, 2587, 1289, 145, 1, 49, 634, 2995, 5678, 4133, 911, 25, 1, 56
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OFFSET
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1,7
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COMMENTS
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Row lengths are 1,1,1,2,2,3,3,4,4,5,5,6,6,...
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LINKS
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FORMULA
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G.f. satisfies A(x,y) = x*y + x*WEIGH(A(x,y)) - x. Shifts up under WEIGH transform.
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EXAMPLE
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Triangle begins:
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;
...
The identity trees with n=6 nodes, as (preorder-) level sequences, together with their number of leaves, and an ASCII rendering, are:
:
: 1: [ 0 1 2 3 4 5 ] 1
: O--o--o--o--o--o
:
: 2: [ 0 1 2 3 4 3 ] 2
: O--o--o--o--o
: .--o
:
: 3: [ 0 1 2 3 4 2 ] 2
: O--o--o--o--o
: .--o
:
: 4: [ 0 1 2 3 4 1 ] 2
: O--o--o--o--o
: .--o
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: 5: [ 0 1 2 3 2 1 ] 3
: O--o--o--o
: .--o
: .--o
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: 6: [ 0 1 2 3 1 2 ] 2
: O--o--o--o
: .--o--o
:
This gives [1, 4, 1], row n=6 of the triangle.
(End)
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PROG
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(PARI)
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)}
A(n)={my(v=[y]); for(n=2, n, v=concat([y], WeighMT(v))); apply(p->Vecrev(p/y), v)}
{ my(T=A(15)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 28 2018
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CROSSREFS
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KEYWORD
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nonn,tabf,eigen
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AUTHOR
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STATUS
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approved
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