A202477 The number of ways to build all endofunctions on each block of every set partition of {1,2,...,n}.

1, 1, 5, 40, 437, 6036, 100657, 1965160, 43937385, 1106488720, 30982333661, 954607270464, 32090625710365, 1168646120904640, 45826588690845705, 1924996299465966976, 86231288506425806033, 4103067277186778016000, 206655307175847710248885

Offset: 0

Author

Geoffrey Critzer, Dec 19 2011

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..385

Crossrefs

Cf. A000262 (the same for permutations), A137452.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(i^(i*j)*b(n-i*j, i-1)*
       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 29 2016

Mathematica

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}] ;
Range[0, nn]! CoefficientList[Series[Exp[t/(1 - t)], {x, 0, nn}], x]

Formula

E.g.f.: exp(T(x)/(1-T(x))) where T(x) is the e.g.f. for A000169.

a(n) ~ n^(n-1/3) * exp(3/2*n^(1/3) - 2/3) / sqrt(3). - Vaclav Kotesovec, Sep 24 2013

a(n) = Sum_{k=0..n} n^(n-k)*binomial(n-1,k-1)*A000262(k). - Fabian Pereyra, Jul 12 2024

The above formula can be written with the Abel polynomials: a(n) = Sum_{k=0..n} (-1)^(n - k) * A137452(n, k) * A000262(k). - Peter Luschny, Jul 13 2024