A178682 The number of functions f:{1,2,...,n}->{1,2,...,n} such that the number of elements that are mapped to m is divisible by m.

1, 1, 2, 5, 13, 42, 150, 576, 2266, 9966, 47466, 237019, 1224703, 6429152, 35842344, 212946552, 1325810173, 8488092454, 55276544436, 362961569008, 2465240278980, 17538501945077, 130454679958312, 1002493810175093, 7838007702606372, 61789072382062638

Offset

0,3

Comments

a(n) is also the number of partitions of n where each block of part i with multiplicity j is marked with a word of length i*j over an n-ary alphabet whose letters appear in alphabetical order and all n letters occur exactly once in the partition. a(3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc. There is a simple bijection between the marked partitions and the functions f. - Alois P. Heinz, Aug 30 2015

Links

Alois P. Heinz, Table of n, a(n) for n = 0..680

Formula

E.g.f.: Product_{j>=1} Sum_{i>=0} x^(j*i)/(j*i)!.

Example

a(3) = 5 because there are 5 such functions: (1,1,1), (1,2,2), (2,1,2), (2,2,1), (3,3,3).
G.f. = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 42*x^5 + 150*x^6 + 576*x^7 + ...

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 30 2015

Mathematica

Range[0, 20]! CoefficientList[Series[Product[Sum[x^(j i)/(j i)!, {i, 0, 20}], {j, 1, 20}], {x, 0, 20}], x]

Prog

(PARI) m=30; my(x='x+O('x^m)); Vec(serlaplace(prod(j=1, m, sum(k=0, m, x^(k*j)/(k*j)!)))) \\ G. C. Greubel, Jan 26 2019
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&*[(&+[x^(k*j)/Factorial(k*j): k in [0..m]]): j in [1..m]]) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jan 26 2019
(Sage) m = 30; T = taylor(product(sum(x^(k*j)/factorial(k*j) for k in (0..m)) for j in (1..m)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jan 26 2019

Crossrefs

Cf. A000110, A005651, A007837, A261774.

Main diagonal of A326500, A326616, A326617.

Row sums of A364285, A364310.

Sequence in context: A149873 A149874 A114297 * A229161 A192745 A299430

Adjacent sequences: A178679 A178680 A178681 * A178683 A178684 A178685

Keyword

nonn,nice

Author

Geoffrey Critzer, Dec 25 2010

Extensions

a(21)-a(25) from Alois P. Heinz, Aug 30 2015

Status

approved