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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. 9
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 (list; graph; refs; listen; history; text; internal format)
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. A005651, A007837, A261774.

Main diagonal of A326500, A326616, A326617.

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

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Last modified December 1 18:36 EST 2022. Contains 358475 sequences. (Running on oeis4.)