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A325201
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Square array whose entry A(n,k) is the number of labeled rooted trees on a set of size n where each node has at most k neighbors that are further away from the root than the node itself, for n >= 0, k >= 0, read by descending antidiagonals.
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6
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0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 6, 0, 0, 1, 2, 9, 24, 0, 0, 1, 2, 9, 60, 120, 0, 0, 1, 2, 9, 64, 540, 720, 0, 0, 1, 2, 9, 64, 620, 6120, 5040, 0, 0, 1, 2, 9, 64, 625, 7620, 83790, 40320, 0, 0, 1, 2, 9, 64, 625, 7770, 113610, 1345680, 362880, 0, 0, 1, 2, 9, 64, 625, 7776, 117390, 1992480, 24811920, 3628800, 0
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OFFSET
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1,9
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COMMENTS
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A preimage constraint on a function is a set of nonnegative integers such that the size of the inverse image of any element is one of the values in that set. View a labeled rooted tree as an endofunction on the set {1,2,...,n} by sending every non-root node to its neighbor that is closer to the root and sending the root to itself.
Thus, A(n,k) is the number of endofunctions on a set of size n with exactly one cyclic point and such that each preimage has at most k entries.
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LINKS
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FORMULA
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A(n,k) = (n-1)! * [x^(n-1)] e_k(x)^n, where e_k(x) is the truncated exponential 1 + x + x^2/2! + ... + x^k/k!. When k>1, the link above yields explicit constants c_k, r_k so that the columns are asymptotically c_k * n^(-3/2) * r_k^-n. Stirling's approximation gives column k=1, and column k=0 is 0.
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EXAMPLE
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Array begins:
0 0 0 0 0 ...
0 1 1 1 1 ...
0 2 2 2 2 ...
0 6 9 9 9 ...
0 24 60 64 64 ...
0 120 540 620 625 ...
0 720 6120 7620 7770 ...
0 5040 83790 113610 117390 ...
0 40320 1345680 1992480 2088520 ...
0 362880 24811920 40194000 42771960 ...
0 3628800 516650400 916927200 991090800 ...
0 39916800 11992503600 23341071600 25635767850 ...
...
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MATHEMATICA
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e[k_][x_] := Sum[x^j/j!, {j, 0, k}];
A[0, _] = A[_, 0] = 0; A[n_, k_] := (n-1)! Coefficient[e[k][x]^n, x, n-1];
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PROG
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(Python)
# print first num_entries entries in column k
import math, sympy; x=sympy.symbols('x')
k=5; num_entries = 64
P=range(k+1); eP=sum([x**d/math.factorial(d) for d in P]); r = [0, 1]; curr_pow = eP
for term in range(1, num_entries-1):
...curr_pow=(curr_pow*eP).expand()
...r.append(curr_pow.coeff(x**term)*math.factorial(term))
print(r)
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CROSSREFS
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Column 1 is A000142, except at n=0 term.
Similar array for arbitrary endofunctions (without limitation on the number of cyclic points) with the same preimage condition {i>=0 | i<=k}: A306800.
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KEYWORD
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AUTHOR
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STATUS
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approved
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