A365061 a(n) is the number of endofunctions on an n-set where there is a single element with a preimage of maximum cardinality.
1, 2, 21, 196, 2105, 27636, 451003, 8938056, 207358929, 5451691060, 158802143621, 5051104945272, 173783789845861, 6424902913267216, 253983495283150095, 10692693172088104336, 477787129703211313697, 22591854186020941025268, 1127404525137567577764013
Offset: 1
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..340
- P. L. Krapivsky, Random Maps with Sociological Flavor, arXiv:2309.08834 [math.CO], 2023. See p. 12.
Programs
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Maple
a:= proc(m) option remember; m*add(binomial(m, j)* b(m-j, min(j-1, m-j), m-1), j=1..m) end: b:= proc(n, i, t) option remember; `if`(n=0, 1, add( b(n-j, i, t-1) *binomial(n-1, j-1)*t, j=1..min(n, i))) end: seq(a(n), n=1..20); # Alois P. Heinz, Aug 25 2023 -
Mathematica
seriesCoeff[n_, b_] := seriesCoeff[n, b] = SeriesCoefficient[(Exp[z]*Gamma[b, z]/Gamma[b])^(n - 1), {z, 0, n - b}]; a[n_] := n*Total[Table[Binomial[n, b]*(n - b)!*seriesCoeff[n, b], {b, 1, n}]]; Monitor[Table[a[n], {n, 1, 19}], {n - 1, a[n - 1]}] (* Robert P. P. McKone, Aug 26 2023 *) -
Maxima
a(n):=n*sum(binomial(n,b)*(n-b)!*coeff(taylor((exp(z)* gamma_incomplete_regularized(b,z))^(n-1),z,0,n),z,n-b),b,1,n);
Formula
a(n) = n*Sum_{b=1..n} binomial(n,b)*(n-b)!*[z^(n-b)](e^z*Gamma(b,z)/Gamma(b))^(n-1).
a(n) mod 2 = A000035(n). - Alois P. Heinz, Aug 25 2023
Extensions
a(16)-a(19) from Alois P. Heinz, Aug 25 2023