OFFSET
1,5
COMMENTS
The lowering (or delta) operator for these polynomials is L = -1 + exp{ 2 + W[ -exp(-2) * (2+D) ] } = Sum_{j >= 1} A074059(j) * D^j / j!.
The raising operator is R = -x { 1 + W[ -exp(-2) * (2+D) ] } = x { 1 + Sum_{j >= 1} (-1)^j * PW(j-1,-2) * D^j / j! }, where PW(j-1,x) are the polynomials of A042977.
W(x) here is W_-1 in the Monir reference and, about x = 0, W[ -exp(-2) * (2+x) ] = -[ 2 + Sum_{j >= 1} (-1)^j * PW(j-1,-2) * x^j / j! ].
From the relation between the delta and raising operators for associated binomial-type polynomials, A074059 = (1,1,2,7,34,...) and S = (1,-PW(0,-2),PW(1,-2),-PW(2,-2),...) = (1, -1, 0, -1, -2, -13, -74, -593, -5298, ...) form a list partition transform pair (see A133314); i.e., S and A074059 have reciprocal e.g.f.s and satisfy mutual recursion relations. Applying Faa di Bruno's formula to L gives other interesting integer relations between S and A074059.
The Bell transform of (-1)^n*factorial(n-1) if n>0, else 1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016
LINKS
F. Chapeau-Blondeau and A. Monir, Numerical Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2, IEEE Trans. on Signal Processing, Vol. 50, No. 9, Sept. 2002, p. 2160-2164.
FORMULA
The row polynomials P(n,t) = Sum_{j=1..n} C(n,j) * t^j satisfy exp[P(.,t) * x] = exp{ -t * [(1+x) * log(1+x) - 2*x] }, with P(0,t) = 1 and [ P(.,x) + P(.,y) ]^n = P(n,x+y). Here, as in the e.g.f., the umbral maneuver P(.,t)^n = P(n,t) is assumed. See Mathworld and Wikipedia on Sheffer sequences and umbral calculus for other general formulas, including expansion theorems.
From Peter Bala, Dec 09 2011: (Start)
E.g.f.: exp(t*(2*x-(1+x)*log(1+x))) = 1 + t*x + (t^2-t)*x^2/2! + (t^3-3*t^2+t)*x^3/3! + ... (Restatement of Copeland's e.g.f. above in umbral notation with P(.,t)^n = P(n,t).).
If a triangular array has an e.g.f. of the form exp(t*F(x)) with F(0) = 0, then the o.g.f.'s for the diagonals of the triangle are rational functions in t (see the Bala link). The rational functions are the coefficients in the compositional inverse (with respect to x) (x-t*F(x))^(-1). In this case (x-t*(2*x-(1+x)*log(1+x)))^(-1) = x/(1-t) - t/(1-t)^3*x^2/2! + (t+2*t^2)/(1-t)^5*x^3/3! - (2*t+6*t^2+7*t^3)/(1-t)^7*x^4/4! + ... . So, for example, the (unsigned) third subdiagonal has o.g.f. (2*t+6*t^2+7*t^3)/(1-t)^7 = 2*t + 20*t^2 + 105*t^3 + 385*t^4 + ... .
(End)
EXAMPLE
Triangle read by rows:
1;
-1, 1;
1, -3, 1;
-2, 7, -6, 1;
6, -20, 25, -10, 1;
-24, 76, -105, 65, -15, 1;
120, -364, 511, -385, 140, -21, 1;
-720, 2108, -2940, 2401, -1120, 266, -28, 1;
...
From R. J. Mathar, Mar 22 2013: (Start)
The matrix inverse starts:
1;
1, 1;
2, 3, 1;
7, 11, 6, 1;
34, 55, 35, 10, 1;
213, 349, 240, 85, 15, 1;
1630, 2695, 1939, 770, 175, 21, 1;
... (End)
MAPLE
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> `if`(n=0, 1, (-1)^n*(n-1)!), 9); # Peter Luschny, Jan 27 2016
MATHEMATICA
max = 10; s = Series[Exp[t*(2*x-(1+x)*Log[1+x])], {x, 0, max}, {t, 0, max}] // Normal; c[n_, j_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, j}]*n!; Table[c[n, j], {n, 1, max}, {j, 1, n}] // Flatten (* Jean-François Alcover, Apr 23 2014, after Peter Bala, duplicate of Copeland's e.g.f. *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[Function[n, If[n == 0, 1, (-1)^n (n-1)!]], rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, after Peter Luschny *)
PROG
(Sage) # uses[bell_matrix from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n*factorial(n-1) if n>0 else 1, 10) # Peter Luschny, Jan 18 2016
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Tom Copeland, Feb 15 2008
EXTENSIONS
More terms from Jean-François Alcover, Apr 23 2014
STATUS
approved