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
G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics, arXiv:quant-ph/0401126, 2004.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
Bruce E. Sagan, A note on Abel polynomials and rooted labeled forests. Discrete Mathematics 44(3): 293-298 (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 [math-ph], 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)^(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) # 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
Row sums are A000248.
KEYWORD
AUTHOR
N. J. A. Sloane, Jan 25 2001
STATUS
approved