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A206823 Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} with exactly k elements x such that |f^(-1)(x)| = 1; n>=0, 0<=k<=n. 3
1, 0, 1, 2, 0, 2, 3, 18, 0, 6, 40, 48, 144, 0, 24, 205, 1000, 600, 1200, 0, 120, 2556, 7380, 18000, 7200, 10800, 0, 720, 24409, 125244, 180810, 294000, 88200, 105840, 0, 5040, 347712, 1562176, 4007808, 3857280, 4704000, 1128960, 1128960, 0, 40320 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums = n^n, all functions f:{1,2,...,n}->{1,2,...,n}.

T(n,n)= n!, bijections on {1,2,...,n}.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..140, flattened

FORMULA

E.g.f.: sum(k=0..n, T(n,k) * y^k * x^n / n! ) = (exp(x) - x + y*x)^n.

EXAMPLE

1

0     1

2     0     2

3     18    0     6

40    48    144   0     24

205   1000  600   1200  0     120

MAPLE

with(combinat): C:= binomial:

b:= proc(t, i, u) option remember; `if`(t=0, 1,

      `if`(i<2, 0, b(t, i-1, u) +add(multinomial(t, t-i*j, i$j)

      *b(t-i*j, i-1, u-j)*u!/(u-j)!/j!, j=1..t/i)))

    end:

T:= (n, k)-> C(n, k)*C(n, k)*k! *b(n-k$2, n-k):

seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Nov 13 2013

MATHEMATICA

nn = 8; Prepend[CoefficientList[Table[n! Coefficient[Series[(Exp[x] - x + y x)^n, {x, 0, nn}], x^n], {n, 1, nn}], y], {1}] // Flatten

CROSSREFS

Row sums give: A000312.

Column k=0 gives: A231797.

Cf. A231602.

Sequence in context: A071547 A220222 A089839 * A151668 A086151 A099040

Adjacent sequences:  A206820 A206821 A206822 * A206824 A206825 A206826

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Feb 12 2012

STATUS

approved

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Last modified October 22 09:57 EDT 2018. Contains 316433 sequences. (Running on oeis4.)