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A292087
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Limit of the number of (unlabeled) rooted trees without unary nodes where n is the difference between the number of leafs and the maximal outdegree as the tree size increases.
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2
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1, 2, 7, 23, 78, 262, 893, 3040, 10411, 35724, 122950, 424004, 1465254, 5071981, 17584226, 61046464, 212197118, 738422362, 2572261241, 8968726829, 31298189180, 109307655964, 382031357974, 1336107044159, 4675807680776, 16372936282017, 57363325974309
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OFFSET
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0,2
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LINKS
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EXAMPLE
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: a(0) = 1:
: o
: //( )\\
: N N N N N N
:
: a(1) = 2:
: o o
: / \ / /|\ \
: o N o N N N N
: / /|\ \ ( )
: N N N N N N N
:
: a(2) = 7:
: o o o o
: / \ / \ /( )\ / | \
: o N o N o N N N o N N
: / \ /( )\ / \ /( )\
: o N o N N N o N N N N N
: /( )\ ( ) ( )
: N N N N N N N N
:
: o o o
: / \ /( )\ / ( \ \
: o o o N N N o o N N
: /( )\ ( ) /|\ ( ) ( )
: N N N N N N N N N N N N N
:
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MAPLE
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b:= proc(n, i, v, k) option remember; `if`(n=0,
`if`(v=0, 1, 0), `if`(i<1 or v<1 or n<v, 0,
`if`(v=n, 1, add(binomial(A(i, k)+j-1, j)*
b(n-i*j, i-1, v-j, k), j=0..min(n/i, v)))))
end:
A:= proc(n, k) option remember; `if`(n<2, n,
add(b(n, n+1-j, j, k), j=2..min(n, k)))
end:
a:= n-> A(2*n+3, n+3)-A(2*n+3, n+2):
seq(a(n), n=0..23);
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MATHEMATICA
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b[n_, i_, v_, k_] := b[n, i, v, k] = If[n == 0,
If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0,
If[v == n, 1, Sum[Binomial[A[i, k] + j - 1, j]*
b[n - i*j, i - 1, v - j, k], {j, 0, Min[n/i, v]}]]]];
A[n_, k_] := A[n, k] = If[n < 2, n,
Sum[b[n, n + 1 - j, j, k], {j, 2, Min[n, k]}]];
a[n_] := A[2*n + 3, n + 3] - A[2*n + 3, n + 2];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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