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A126285
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Number of partial mappings (or mapping patterns) from n points to themselves; number of partial endofunctions.
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6
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1, 2, 6, 16, 45, 121, 338, 929, 2598, 7261, 20453, 57738, 163799, 465778, 1328697, 3798473, 10883314, 31237935, 89812975, 258595806, 745563123, 2152093734, 6218854285, 17988163439, 52078267380, 150899028305, 437571778542, 1269754686051, 3687025215421
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OFFSET
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0,2
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COMMENTS
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If an endofunction is partial, then some points may be unmapped (or mapped to "undefined").
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LINKS
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FORMULA
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Euler transform of A002861 + A000081 = [1, 2, 4, 9, 20, 51, 125, 329, 862, 2311, ... ] + [ 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, ...] = A124682.
a(n) ~ c * d^n / sqrt(n), where d = 2.95576528565... is the Otter's rooted tree constant (see A051491) and c = 1.309039781943936352117502717... - Vaclav Kotesovec, Mar 29 2014
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MATHEMATICA
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nmax = 28;
a81[n_] := a81[n] = If[n<2, n, Sum[Sum[d*a81[d], {d, Divisors[j]}]*a81[n-j ], {j, 1, n-1}]/(n-1)];
A[n_] := A[n] = If[n<2, n, Sum[DivisorSum[j, #*A[#]&]*A[n-j], {j, 1, n-1} ]/(n-1)];
H[t_] := Sum[A[n]*t^n, {n, 0, nmax+2}];
F = 1/Product[1 - H[x^n], {n, 1, nmax+2}] + O[x]^(nmax+2);
A1372 = CoefficientList[F, x];
a[n_] := Sum[a81[k] * A1372[[n-k+2]], {k, 0, n+1}];
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PROG
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(Sage)
Pol.<t> = InfinitePolynomialRing(QQ)
@cached_function
def Z(n):
if n==0: return Pol.one()
return sum(t[k]*Z(n-k) for k in (1..n))/n
def pmagmas(n, k=1): # number of partial k-magmas on a set of n elements up to isomorphism
P = Z(n)
q = 0
coeffs = P.coefficients()
count = 0
for m in P.monomials():
p = 1
V = m.variables()
T = cartesian_product(k*[V])
for t in T:
r = [Pol.varname_key(str(u))[1] for u in t]
j = [m.degree(u) for u in t]
D = 1
lcm_r = lcm(r)
for d in divisors(lcm_r):
try: D += d*m.degrees()[-d-1]
except: break
p *= D^(prod(r)/lcm_r*prod(j))
q += coeffs[count]*p
count += 1
return q
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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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