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A137452
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Triangular array of the coefficients of the sequence of Abel polynomials A(n,x) := x*(x-n)^(n-1).
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12
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1, 0, 1, 0, -2, 1, 0, 9, -6, 1, 0, -64, 48, -12, 1, 0, 625, -500, 150, -20, 1, 0, -7776, 6480, -2160, 360, -30, 1, 0, 117649, -100842, 36015, -6860, 735, -42, 1, 0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1, 0, 43046721, -38263752, 14880348, -3306744, 459270, -40824, 2268, -72, 1
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OFFSET
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0,5
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COMMENTS
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The Abel polynomials are associated with the Abel operator t*exp(y*t)*p(x) = t*p(x+y).
Abs(T(n,k)) is the number of rooted labeled trees on n+1 vertices with a root degree k (Clarke's formula).
The row sums in the unsigned case, Sum_{k=0..n} abs(T(n,k)), count the trees on n+1 labeled nodes, A000272(n+1). (End)
Exponential Riordan array [1, W(x)], W(x) the Lambert W-function. - Paul Barry, Nov 19 2010
The inverse array is the exponential Riordan array [1, x*exp(x)], which is A059297. - Peter Bala, Apr 08 2013
The inverse Bell transform of [1,2,3,...]. See A264428 for the Bell transform and A264429 for the inverse Bell transform. - Peter Luschny, Dec 20 2015
Also the Bell transform of (-1)^n*(n+1)^n. - Peter Luschny, Jan 18 2016
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REFERENCES
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Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 14 and 29
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LINKS
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FORMULA
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Row n gives the coefficients of the expansion of x*(x-n)^(n-1).
From the exponential Riordan (also Sheffer of Jabotinsky) type (1, LambertW) array (see comments).
E.g.f. of column sequence k, LambertW(x)^k/k!, for k >= 0.
E.g.f. of row polynomials P_n(y) = Sum_{k=0..n} T(n, k)*y^k: exp(y*LambertW(x)).
Recurrence for T: T(n, k) = 0 for n < k; T(n, 0) = 1 for n = 0 otherwise 0; T(n, k) = (n/k)*Sum_{j=0..n-k} binomial(k-1+j,k-1)*(-1)^j*T(n-1, k-1+j). (Jabotinsky type convolution triangle, the e.g.f.s for the a- and z-sequences are exp(-x), and 0. See the link in A006232.)
Recurrence for column k of T: T(n, k) = 0 for n < k, T(k, k) = 1, for k >= 0 otherwise T(n, k) = (n!*k/(n-k))*Sum_{j=k..n-1} (1/j!)*beta(n-1-j)*T(j, k), where beta(n) = A264234(n+1)/A095996(n+1) = {-1, 2, -9/2, 32/3, -625/24, ...} with o.g.f. d/dx(log(LambertW(x)/x). See the Boas-Buck or Rainville references given in A046521, and my Aug 10 2017 comment there.
Recurrence for the row polynomials P_0(x) = 1, and P_n(x) = x*substitute(z=d/dx, exp(-z)/(1+z)) P_(n-1)(x), for n >= 1, with coefficient z^k of exp(-z)/(1+z) given by (-1)^k*A061354(k)/A061355(k). See the Roman reference Corollary 3.7.2., p. 50. (End)
The column sequences for the unsigned triangle Abs(T(n, k)), for k >= 2, are also given by {n^(n-k)*(n-1)*s(k-2, n)/(k-1)!}_{n>=k} with the row polynomials s(n, x) = risingfactorial(x - (n+1), n) of A049444. - Wolfdieter Lang, Nov 21 2022
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EXAMPLE
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1;
0, 1;
0, -2, 1;
0, 9, -6, 1;
0, -64, 48, -12, 1;
0, 625, -500, 150, -20, 1;
0, -7776, 6480, -2160, 360, -30, 1;
0, 117649, -100842, 36015, -6860, 735, -42, 1;
0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1;
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MAPLE
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T := proc(n, k) if n = 0 and k = 0 then 1 else binomial(n-1, k-1)*(-n)^(n-k) fi end; seq(print(seq(T(n, k), k=0..n)), n=0..7); # Peter Luschny, Jan 14 2009
# The function BellMatrix is defined in A264428.
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MATHEMATICA
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a0 = 1 a[x, 0] = 1; a[x, 1] = x; a[x_, n_] := x*(x - a0*n)^(n - 1); Table[Expand[a[x, n]], {n, 0, 10}]; a1 = Table[CoefficientList[a[x, n], x], {n, 0, 10}]; Flatten[a1]
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, (-n-1)^n], rows = 12];
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PROG
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(Sage) # uses[inverse_bell_transform from A264429]
nat = [n for n in (1..dim)]
return inverse_bell_transform(dim, nat)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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