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 A059297 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1. 15
 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 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 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 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: A035543 A105546 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

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Last modified January 25 19:06 EST 2021. Contains 340427 sequences. (Running on oeis4.)