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A135338 Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomial-type) with raising operator -x { 1 + W[ -exp(-2) * (2+D) ] } where W is the Lambert W multi-valued function.

%I A135338 #57 Oct 14 2020 10:35:12
%S A135338 1,-1,1,1,-3,1,-2,7,-6,1,6,-20,25,-10,1,-24,76,-105,65,-15,1,120,-364,
%T A135338 511,-385,140,-21,1,-720,2108,-2940,2401,-1120,266,-28,1,5040,-14328,
%U A135338 19720,-16632,8841,-2772,462,-36,1,-40320,111816,-151620,129340,-73605,27237,-6090,750,-45,1
%N A135338 Triangle read by rows: row n gives coefficients C(n,j) for a Sheffer sequence (binomial-type) with raising operator -x { 1 + W[ -exp(-2) * (2+D) ] } where W is the Lambert W multi-valued function.
%C A135338 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!.
%C A135338 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.
%C A135338 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! ].
%C A135338 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.
%C A135338 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
%H A135338 Peter Bala, <a href="/A112007/a112007_Bala.txt">Diagonals of triangles with generating function exp(t*F(x)).</a>
%H A135338 F. Chapeau-Blondeau and A. Monir, <a href="http://dx.doi.org/10.1109/TSP.2002.801912">Numerical Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2</a>, IEEE Trans. on Signal Processing, Vol. 50, No. 9, Sept. 2002, p. 2160-2164.
%F A135338 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.
%F A135338 From _Peter Bala_, Dec 09 2011: (Start)
%F A135338 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).).
%F A135338 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 + ... .
%F A135338 (End)
%e A135338 Triangle read by rows:
%e A135338      1;
%e A135338     -1,    1;
%e A135338      1,   -3,     1;
%e A135338     -2,    7,    -6,    1;
%e A135338      6,  -20,    25,  -10,     1;
%e A135338    -24,   76,  -105,   65,   -15,   1;
%e A135338    120, -364,   511, -385,   140, -21,   1;
%e A135338   -720, 2108, -2940, 2401, -1120, 266, -28, 1;
%e A135338 ...
%e A135338 From _R. J. Mathar_, Mar 22 2013: (Start)
%e A135338 The matrix inverse starts:
%e A135338      1;
%e A135338      1,    1;
%e A135338      2,    3,    1;
%e A135338      7,   11,    6,   1;
%e A135338     34,   55,   35,  10,   1;
%e A135338    213,  349,  240,  85,  15,  1;
%e A135338   1630, 2695, 1939, 770, 175, 21, 1;
%e A135338   ... (End)
%p A135338 # The function BellMatrix is defined in A264428.
%p A135338 # Adds (1,0,0,0, ..) as column 0.
%p A135338 BellMatrix(n -> `if`(n=0,1,(-1)^n*(n-1)!), 9); # _Peter Luschny_, Jan 27 2016
%t A135338 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. *)
%t A135338 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A135338 rows = 12;
%t A135338 M = BellMatrix[Function[n, If[n == 0, 1, (-1)^n (n-1)!]], rows];
%t A135338 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 26 2018, after _Peter Luschny_ *)
%o A135338 (Sage) # uses[bell_matrix from A264428]
%o A135338 # Adds a column 1,0,0,0, ... at the left side of the triangle.
%o A135338 bell_matrix(lambda n: (-1)^n*factorial(n-1) if n>0 else 1, 10) # _Peter Luschny_, Jan 18 2016
%Y A135338 Cf. A134685, A135494.
%K A135338 sign,tabl
%O A135338 1,5
%A A135338 _Tom Copeland_, Feb 15 2008
%E A135338 More terms from _Jean-François Alcover_, Apr 23 2014