login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A137452 Triangular array of the coefficients of the sequence of Abel polynomials A(n,x) := x*(x-n)^(n-1). 2
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, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums give A177885.

The Abel polynomials are associated with the Abel operator t*exp(y*t)*p(x) = t*p(x+y).

From Peter Luschny, Jan 14 2009: (Start)

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

REFERENCES

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 14 and 29

LINKS

Seiichi Manyama, Rows n = 0..139, flattened

W. Y. Chen, A general bijective algorithm for trees, PNAS December 1, 1990 vol. 87 no. 24 9635-9639.

L. E. Clarke, On Cayley's formula for counting trees, J. London Math. Soc. 33 (1958), 471-475.

Péter L. Erdős and L. A. Székely, Applications of Antilexicographic Order. I., An Enumerative Theory of Trees, Adv. in Appl. Math. 10, (1989) 488-496.

Eric Weisstein's World of Mathematics Abel Polynomial

Wikipedia, Abel Polynomials

Bao-Xuan Zhu, Total positivity from a generalized cycle index polynomial, arXiv:2006.14485 [math.CO], 2020.

FORMULA

Row n gives the coefficients of the expansion of x*(x-n)^(n-1).

Abs(T(n,k)) = Sum_{k=0..n} C(n-1,k-1)*n^(n-k). - Peter Luschny, Jan 14 2009

EXAMPLE

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;

MAPLE

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.

BellMatrix(n -> (-n-1)^n, 9); # Peter Luschny, Jan 27 2016

MATHEMATICA

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];

Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

PROG

(Sage) # uses[inverse_bell_transform from A264429]

def A137452_matrix(dim):

    nat = [n for n in (1..dim)]

    return inverse_bell_transform(dim, nat)

A137452_matrix(10) # Peter Luschny, Dec 20 2015

CROSSREFS

Row sums A177885.

Cf. A000272, A061356, A059297 (inverse array), A264429.

Sequence in context: A246658 A274740 A327350 * A158335 A111595 A021478

Adjacent sequences:  A137449 A137450 A137451 * A137453 A137454 A137455

KEYWORD

tabl,sign

AUTHOR

Roger L. Bagula, Apr 18 2008

EXTENSIONS

Better name by Peter Bala, Apr 08 2013

Edited by Joerg Arndt, Apr 08 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 24 16:24 EDT 2021. Contains 348233 sequences. (Running on oeis4.)