Inverse of the matrix of coefficients of f_k(x)
	*  The matrix of coefficients of f_k(x) is inverted
	*  The columns define then some sequences, which can be found in OEIS
Matrix( f_k(x) )^-1
	g0(x)=	-1	.	.	.	.	.	.	.	.	.	.	.
	g1(x)=	-1	1	.	.	.	.	.	.	.	.	.	.
	g2(x)=	-3	1	2	.	.	.	.	.	.	.	.	.
	g3(x)=	-13	7	3	3	.	.	.	.	.	.	.	.
	g4(x)=	-75	37	28	6	4	.	.	.	.	.	.	.
	g5(x)=	-541	271	185	70	10	5	.	.	.	.	.	.
	g6(x)=	-4683	2341	1626	555	140	15	6	.	.	.	.	.
	g7(x)=	-47293	23647	16387	5691	1295	245	21	7	.	.	.	.
	g8(x)=	-545835	272917	189176	65548	15176	2590	392	28	8	.	.	.
	g9(x)=	-7087261	3543631	2456253	851292	196644	34146	4662	588	36	9	.	.
	g10(x)=	-102247563	51123781	35436310	12281265	2837640	491610	68292	7770	840	45	10	.
	g11(x)=	-1622632573	811316287	562361591	194899705	45031305	7803510	1081542	125202	12210	1155	55	11
	...	...
	OEIS:	A000670	A089677
	a few Observations:
	1	The rowsums are zero (except for the first one)
	2	second column minus tail is 1,-1,1,-1,1,-1
	3	roughly similar column-schemes seem to exist with weighted column-sums
	4	Subdiagonals from column 2 on are binomial multiples of the leading subdiagonal-entry in column 1
	Observation 4:
	g0(x)=	-1	.	.	.	.	.	.	.	.	.	.	.
	g1(x)=	-1	1*1	.	.	.	.	.	.	.	.	.	.
	g2(x)=	-3	1*1	2*1	.	.	.	.	.	.	.	.	.
	g3(x)=	-13	1*7	3*1	3*1	.	.	.	.	.	.	.	.
	g4(x)=	-75	1*37	4*7	6*1	4*1	.	.	.	.	.	.	.
	g5(x)=	-541	1*271	5*37	10*7	10*1	5*1	.	.	.	.	.	.
	g6(x)=	-4683	1*2341	6*271	15*37	20*7	15*1	6*1	.	.	.	.	.
References in OEIS:
	A000670	Number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements.
		1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, 1622632573,
	COMMENT
	1	Number of ways n competitors can rank in a competition, allowing for the possibility of ties.
	2	Also number of asymmetric generalized weak orders on n points.
	3	Also called the ordered Bell numbers, i.e. the number of ordered partitions of {1,..,n}.
	4	A weak order is a relation that is transitive and complete.
	5	Called Fubini numbers by Comtet: counts formulae in Fubini theorem when switching the order of summation in multiple sums. - Olivier Gerard, Sep 30, 2002
	6	If the points are unlabeled then the answer is a(0) = 1, a(n) = 2^(n-1) (cf. A011782).
	7	For n>0, a(n) is the number of elements in the Coxeter complex of type A_{n-1}. The corresponding sequence for type B is A080253 and there one can find a worked example as well as a geometric interpretation. - Tim Honeywill & Paul Boddington (tch(AT)maths.warwick.ac.uk), Feb 10 2003
	8	Also number of labeled (1+2)-free posets. - Detlef Pauly, May 25 2003
	9	Also the number of chains of subsets starting with the empty set and ending with a set of n distinct objects. - Andy Niedermaier (aniedermaier(AT)hmc.edu), Feb 20 2004
	10	Stirling transform of A007680(n) = [3, 10, 42, 216, . . . ] gives [3,13,75,541,...]. - Michael Somos Mar 04 2004
	11	Stirling transform of a(n)=[1,3,13,75,...] is A083355(n)=[1,4,23,175,...]. - Michael Somos Mar 04 2004
	12	Stirling transform of A000142(n)=[1,2,6,24,120,...] is a(n)=[1,3,13,75,...]. - Michael Somos Mar 04 2004
	13	Stirling transform of A005359(n-1)=[1,0,2,0,24,0,...] is a(n-1)=[1,1,3,13,75,...]. - Michael Somos Mar 04 2004
	14	Stirling transform of A005212(n-1)=[0,1,0,6,0,120,0,...] is a(n-1)=[0,1,3,13,75,...]. - Michael Somos Mar 04 2004
	15	Unreduced denominators in convergent to log(2) = lim[n->inf, na(n-1)/a(n)].
	16	a(n) congruent a(n+(p-1)p^(h-1)) (mod p^h) for n>=h (see Barsky).
	17	Stirling-Bernoulli transform of 1/(1-x^2). - Paul Barry (pbarry(AT)wit.ie), Apr 20 2005
	18	This is the sequence of moments of the probability distribution of the number of tails before the first head in a sequence of fair coin tosses. The sequence of cumulants of the same probability distribution is A000629. That sequence is 2 times the result of deletion of the first term of this sequence. - Michael Hardy (hardy(AT)math.umn.edu), May 01 2005
	19	With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j,i) = the j-th part of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i, and prod_{j=1}^{d(i)} = product over j one has: a(n) = sum_{i=1}^{p(n)} (n!/(prod_{j=1}^{p(i)}p(i,j)!)) * (p(i)!/(prod_{j=1}^{d(i)} m(i,j)!)) - Thomas Wieder (wieder.thomas(AT)t-online.de), May 18 2005
	20	The number of chains among subsets of [n]. The summed term in the new formula is the number of such chains of length k. - Micha Hofri (hofri(AT)wpi.edu), Jul 01 2006
	A089677	Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.
		0, 1, 1, 7, 37, 271, 2341, 23647, 272917, 3543631, 51123781, 811316287,
	COMMENT
	1	Stirling transform of A005212(n)=[1,0,6,0,120,0,5040,...] is a(n)=[1,1,7,37,271,...]. - Michael Somos Mar 04 2004
	FORMULA
	1	E.g.f.	(exp(x)-1)/(exp(x)*(2-exp(x))). a(n)=Sum(Binomial(n, k)(-1)^(n-k)Sum(i! Stirling2(k, i), i=1, ..k), k=0, .., n).
	2	MATHEM'CA	Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}]
	3	PARI/GP	a(n)=if(n<0, 0, n!*polcoeff(subst(y/(1-y^2), y, exp(x+x*O(x^n))-1), n))