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

1,2

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.

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

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 *)

Prog

(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);

Crossrefs

Cf. A000035, A000312 (endofunctions), A351118.

Sequence in context: A037756 A037644 A329553 * A110253 A185634 A077249

Adjacent sequences: A365058 A365059 A365060 * A365062 A365063 A365064

Keyword

nonn

Author

Aaron O. Schweiger, Aug 19 2023

Extensions

a(16)-a(19) from Alois P. Heinz, Aug 25 2023

Status

approved