

A059297


Triangle of idempotent numbers binomial(n,k)*k^(nk), version 1.


16



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
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OFFSET

0,5


COMMENTS

T(n,k) = C(n,k)*k^(nk) 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 nk 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^(nk) ways to map nk 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
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, Oneparameter groups and combinatorial physics, arXiv:quantph/0401126, 2004.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495517.
Bruce E. Sagan, A note on Abel polynomials and rooted labeled forests. Discrete Mathematics 44(3): 293298 (1983).
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 96, [Tom Copeland, Dec 20 2018].
Eric Weisstein's World of Mathematics, Abel Polynomial
Eric Weisstein's World of Mathematics, Idempotent Number
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [mathph], 2021.


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)^(k1), 0 <= k <= n. O.g.f. for the kth column: A(k,1/x) = x^k/(1k*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) (xt*x*exp(x))^1 = x/(1t) + 2*t/(1t)^3*x^2/2! + (3*t+9*t^2)/(1t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1t)^7*x^4/4! + .... For example, the o.g.f. for second subdiagonal is (3*t+9*t^2)/(1t)^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..nk} (j+1)*binomial(n,j)*T(nj,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/(12*x)^3 = x^2+6*x^3+24*x^4+80*x^5+....


MAPLE

T:= (n, k)> binomial(n, k) *k^(nk):
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^(nk): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015
(Sage) # uses[bell_transform from 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, A059298, A059299, 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: A105546 A342237 A339030 * A267222 A077874 A286274
Adjacent sequences: A059294 A059295 A059296 * A059298 A059299 A059300


KEYWORD

nonn,easy,tabl


AUTHOR

N. J. A. Sloane, Jan 25 2001


STATUS

approved



