This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A137452 #110 Feb 16 2025 08:33:07
%S A137452 1,0,1,0,-2,1,0,9,-6,1,0,-64,48,-12,1,0,625,-500,150,-20,1,0,-7776,
%T A137452 6480,-2160,360,-30,1,0,117649,-100842,36015,-6860,735,-42,1,0,
%U A137452 -2097152,1835008,-688128,143360,-17920,1344,-56,1,0,43046721,-38263752,14880348,-3306744,459270,-40824,2268,-72,1
%N A137452 Triangular array of the coefficients of the sequence of Abel polynomials A(n,x) := x*(x-n)^(n-1).
%C A137452 Row sums give A177885.
%C A137452 The Abel polynomials are associated with the Abel operator t*exp(y*t)*p(x) = t*p(x+y).
%C A137452 From _Peter Luschny_, Jan 14 2009: (Start)
%C A137452 Abs(T(n,k)) is the number of rooted labeled trees on n+1 vertices with a root degree k (Clarke's formula).
%C A137452 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)
%C A137452 Exponential Riordan array [1, W(x)], W(x) the Lambert W-function. - _Paul Barry_, Nov 19 2010
%C A137452 The inverse array is the exponential Riordan array [1, x*exp(x)], which is A059297. - _Peter Bala_, Apr 08 2013
%C A137452 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
%C A137452 Also the Bell transform of (-1)^n*(n+1)^n. - _Peter Luschny_, Jan 18 2016
%D A137452 Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 14 and 29
%H A137452 Seiichi Manyama, <a href="/A137452/b137452.txt">Rows n = 0..139, flattened</a>
%H A137452 W. Y. Chen, <a href="https://doi.org/10.1073/pnas.87.24.9635">A general bijective algorithm for trees</a>, PNAS December 1, 1990 vol. 87 no. 24 9635-9639.
%H A137452 L. E. Clarke, <a href="https://doi.org/10.1112/jlms/s1-33.4.471">On Cayley's formula for counting trees</a>, J. London Math. Soc. 33 (1958), 471-475.
%H A137452 Péter L. Erdős and L. A. Székely, <a href="http://www.renyi.hu/~elp/Enumeration/ErdosSzekely-AAM89.pdf">Applications of Antilexicographic Order. I., An Enumerative Theory of Trees</a>, Adv. in Appl. Math. 10, (1989) 488-496.
%H A137452 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AbelPolynomial.html">Abel Polynomial</a>.
%H A137452 Wikipedia, <a href="https://en.wikipedia.org/wiki/Abel_polynomials">Abel Polynomials</a>.
%H A137452 Bao-Xuan Zhu, <a href="https://arxiv.org/abs/2006.14485">Total positivity from a generalized cycle index polynomial</a>, arXiv:2006.14485 [math.CO], 2020.
%F A137452 Row n gives the coefficients of the expansion of x*(x-n)^(n-1).
%F A137452 Abs(T(n,k)) = C(n-1,k-1)*n^(n-k). - _Peter Luschny_, Jan 14 2009
%F A137452 From _Wolfdieter Lang_, Nov 08 2022: (Start)
%F A137452 From the exponential Riordan (also Sheffer of Jabotinsky) type (1, LambertW) array (see comments).
%F A137452 E.g.f. of column sequence k, LambertW(x)^k/k!, for k >= 0.
%F A137452 E.g.f. of row polynomials P_n(y) = Sum_{k=0..n} T(n, k)*y^k: exp(y*LambertW(x)).
%F A137452 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.)
%F A137452 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.
%F A137452 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)
%F A137452 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
%e A137452 Triangle begins:
%e A137452 1;
%e A137452 0, 1;
%e A137452 0, -2, 1;
%e A137452 0, 9, -6, 1;
%e A137452 0, -64, 48, -12, 1;
%e A137452 0, 625, -500, 150, -20, 1;
%e A137452 0, -7776, 6480, -2160, 360, -30, 1;
%e A137452 0, 117649, -100842, 36015, -6860, 735, -42, 1;
%e A137452 0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1;
%p A137452 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
%p A137452 # The function BellMatrix is defined in A264428.
%p A137452 BellMatrix(n -> (-n-1)^n, 9); # _Peter Luschny_, Jan 27 2016
%t A137452 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]
%t A137452 (* Second program: *)
%t A137452 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A137452 B = BellMatrix[Function[n, (-n-1)^n], rows = 12];
%t A137452 Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A137452 (Sage) # uses[inverse_bell_transform from A264429]
%o A137452 def A137452_matrix(dim):
%o A137452 nat = [n for n in (1..dim)]
%o A137452 return inverse_bell_transform(dim, nat)
%o A137452 A137452_matrix(10) # _Peter Luschny_, Dec 20 2015
%Y A137452 Row sums A177885.
%Y A137452 Cf. A000272, A061356, A059297 (inverse array), A264429.
%Y A137452 Cf. A061354, A095996, A264234, A061355, A049444.
%K A137452 tabl,sign,easy
%O A137452 0,5
%A A137452 _Roger L. Bagula_, Apr 18 2008
%E A137452 Better name by _Peter Bala_, Apr 08 2013
%E A137452 Edited by _Joerg Arndt_, Apr 08 2013