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A291662
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Number of ordered rooted trees with 2n non-root nodes such that the maximal outdegree equals n.
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3
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1, 1, 8, 53, 326, 1997, 12370, 77513, 490306, 3124541, 20030000, 129024469, 834451788, 5414950283, 35240152706, 229911617041, 1503232609082, 9847379391133, 64617565719052, 424655979547781, 2794563003870310, 18412956934908669, 121455445321173578
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OFFSET
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0,3
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COMMENTS
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a(n) is also the number of permutations p of [2n] such that in 0p the largest up-jump equals n and no down-jump is larger than 1. An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here. a(2) = 8: 1243, 1324, 1342, 2134, 2143, 2314, 2341, 2431.
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LINKS
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FORMULA
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EXAMPLE
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a(2) = 8:
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MATHEMATICA
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b[n_, t_, k_] := b[n, t, k] = If[n == 0, 1, If[t > 0, Sum[b[j - 1, k, k]* b[n - j, t - 1, k], {j, 1, n}], b[n - 1, k, k]]];
T[n_, k_] := b[n, k - 1, k - 1] - If[k == 1, 0, b[n, k - 2, k - 2]];
a[n_] := T[2n, n];
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PROG
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(Python)
from sympy.core.cache import cacheit
@cacheit
def b(u, o, k): return 1 if u + o==0 else sum([b(u - j, o + j - 1, k) for j in range(1, min(1, u) + 1)]) + sum([b(u + j - 1, o - j, k) for j in range(1, min(k, o) + 1)])
def a(n): return b(0, 2*n, n) - (0 if n==0 else b(0, 2*n, n - 1))
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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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