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A059297 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1. 14
1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 12, 1, 0, 5, 80, 90, 20, 1, 0, 6, 240, 540, 240, 30, 1, 0, 7, 672, 2835, 2240, 525, 42, 1, 0, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 0, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 0, 10, 11520, 262440 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

T(n,k) = C(n,k)*k^(n-k) is the number of functions f from domain [n] to codomain [n+1] such that f(x)=n+1 for exactly k elements x of [n] and f(f(x))=n+1 for the remaining n-k elements x of [n]. Subsequently, row sums of T(n,k) provide the number of functions f:[n]->[n+1] such that either f(x)=n+1 or f(f(x))=n+1 for every x in [n]. We note that there are C(n,k) ways to choose the k elements mapped to n+1 and there are k^(n-k) ways to map n-k elements to a set of k elements. - Dennis P. Walsh, Sep 05 2012

Conjecture: the matrix inverse is A137452. - R. J. Mathar, Mar 12 2013

The above conjecture is correct. This triangle is the exponential Riordan array [1, x*exp(x)]. Thus the  inverse array is the exponential Riordan array [ 1, W(x)], which equals A137452. - Peter Bala, Apr 08 2013

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

P. Bala, Diagonals of triangles with generating function exp(t*F(x)).

G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics, arXiv:quant-ph/0401126, 2004.

Bruce E. Sagan, A note on Abel polynomials and rooted labeled forests. Discrete Mathematics 44(3): 293-298 (1983)

Eric Weisstein's World of Mathematics, Abel Polynomial

Eric Weisstein's World of Mathematics, Idempotent Number

FORMULA

E.g.f.: exp(x*y*exp(y)). - Vladeta Jovovic, Nov 18 2003

Up to signs, this is the triangle of connection constants expressing the monomials x^n as a linear combination of the Abel polynomials A(k,x) := x*(x+k)^(k-1), 0 <= k <= n. O.g.f. for the k-th column: A(-k,1/x) = x^k/(1-k*x)^(k+1). Cf. A061356. Examples are given below. - Peter Bala, Oct 09 2011

The o.g.f.'s for the diagonals of this triangle are the rational functions occurring in the expansion of the compositional inverse (with respect to x) (x-t*x*exp(x))^-1 = x/(1-t) + 2*t/(1-t)^3*x^2/2! + (3*t+9*t^2)/(1-t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1-t)^7*x^4/4! + .... For example, the o.g.f. for second subdiagonal is (3*t+9*t^2)/(1-t)^5 = 3*t + 24*t^2 + 90*t^3 + 240*t^4 + .... See the Bala link. The coefficients of the numerator polynomials are listed in A202017. - Peter Bala, Dec 08 2011

Recurrence equation: T(n+1,k+1) = Sum_{j=0..n-k} (j+1)*binomial(n,j)*T(n-j,k). - Peter Bala, Jan 13 2015

The Bell transform of [1,2,3,...]. See A264428 for the Bell transform. - Peter Luschny, Dec 20 2015

EXAMPLE

Triangle begins:

1;

0,  1;

0,  2,   1;

0,  3,   6,    1;

0,  4,  24,   12,    1;

0,  5,  80,   90,   20,   1;

0,  6, 240,  540,  240,  30,  1;

0,  7, 672, 2835, 2240, 525, 42,  1;

Row 4. Expansion of x^4 in terms of Abel polynomials:

x^4 = -4*x+24*x*(x+2)-12*x*(x+3)^2+x*(x+4)^3.

O.g.f. for column 2: A(-2,1/x) = x^2/(1-2*x)^3 = x^2+6*x^3+24*x^4+80*x^5+....

MAPLE

T:= (n, k)-> binomial(n, k) *k^(n-k):

seq (seq (T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 05 2012

MATHEMATICA

nn=10; f[list_]:=Select[list, #>0&]; Prepend[Map[Prepend[#, 0]&, Rest[Map[f, Range[0, nn]!CoefficientList[Series[Exp[y x Exp[x]], {x, 0, nn}], {x, y}]]]], {1}]//Grid  (* Geoffrey Critzer, Feb 09 2013 *)

t[n_, k_] := Binomial[n, k]*k^(n - k); Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

PROG

(MAGMA) /* As triangle */ [[Binomial(n, k)*k^(n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

(Sage)

# The function bell_transform is defined in A264428.

def A059297_row(n):

    nat = [k for k in (1..n)]

    return bell_transform(n, nat)

[A059297_row(n)  for n in range(8)] # Peter Luschny, Dec 20 2015

CROSSREFS

There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248.

Cf. A061356, A202017, A137452 (inverse array), A264428.

Sequence in context: A202178 A035543 A105546 * A267222 A077874 A230360

Adjacent sequences:  A059294 A059295 A059296 * A059298 A059299 A059300

KEYWORD

nonn,easy,tabl

AUTHOR

N. J. A. Sloane, Jan 25 2001

STATUS

approved

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Last modified March 27 08:41 EDT 2017. Contains 284146 sequences.