A002050 -id:A002050 - cp's OEIS frontend

A052856 E.g.f.: (1-3exp(x)+exp(2x))/(exp(x)-2).

1, 2, 4, 14, 76, 542, 4684, 47294, 545836, 7087262, 102247564, 1622632574, 28091567596, 526858348382, 10641342970444, 230283190977854, 5315654681981356, 130370767029135902, 3385534663256845324

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.
Stirling transform of A005212(n-1)=[1,1,0,6,0,120,0,...] is a(n-1)=[1,2,4,14,76,...]. - Michael Somos, Mar 04 2004
Stirling transform of (-1)^n*A052612(n-1)=[0,2,-2,12,-24,...] is a(n-1)=[0,2,4,14,76,...]. - Michael Somos, Mar 04 2004
Stirling transform of A000142(n)=[2,2,6,24,120,...] is a(n)=[2,2,4,14,76,...]. - Michael Somos, Mar 04 2004

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 824

A000670(n)=a(n)-1, if n>0. A032109(n)=a(n)/2, if n>0.

A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012

spec := [S,{B=Sequence(C),C=Set(Z,1 <= card),S=Union(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(1-3Exp[x]+Exp[x]^2)/(-2+Exp[x]),{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Nov 24 2012 *)

a(n)=if(n<0,0,n!*polcoeff(subst(y+1/(1-y),y,exp(x+x*O(x^n))-1),n))

E.g.f.: (1-3*exp(x)+exp(x)^2)/(-2+exp(x))

a(n) ~ n!/(2*(log(2))^(n+1)). - Vaclav Kotesovec, Oct 05 2013

New name using e.g.f., Vaclav Kotesovec, Oct 05 2013

A076726 a(n) = Sum_{k>=0} k^n/2^k.

Original entry on oeis.org

2, 2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522, 204495126, 3245265146, 56183135190, 1053716696762, 21282685940886, 460566381955706, 10631309363962710, 260741534058271802, 6771069326513690646

Offset: 0

Charles G. Waldman (cgw(AT)alum.mit.edu), Oct 27 2002

nonn

			a(0) = 2 because 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 2; a(1) = 2 because 0 + 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + ... = 2.
G.f. = 2 + 2*x + 6*x^2 + 26*x^3 + 150*x^4 + 1082*x^5 + 9366*x^6 + 94586*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Olivier Golinelli, Remote control system of a binary tree of switches - II. balancing for a perfect binary tree, arXiv:2405.16968 [cs.DM], 2024. See p. 17.
Ramesh L. Srigiriraju, Recurrences for A076726

Same as A000629 except for a(0).

A000629, A000670, A002050, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012

a[n_] := Sum[(k^n)/(2^k), {k, 0, Infinity}]; Table[ a[n], {n, 0, 18}]
a[n_] := (-1)^(n+1) PolyLog[-n, 2] (* Vladimir Reshetnikov, Jan 23 2011 *)

a(n)=abs(polylog(-n, 2)) \\ Charles R Greathouse IV, Jul 15 2014

a(n) = 2*A000670(n). - Philippe Deléham, Mar 06 2004
a(n) ~ n! / (log(2))^(n+1). - Vaclav Kotesovec, Nov 28 2013
From Jianing Song, May 04 2022: (Start)
a(0) = 2, a(n) = Sum_{k=0..n-1} binomial(n,k)*a(k) for n >= 1.
G.f.: Sum_{k>=0} 1/(2^k*(1-k*x)).
E.g.f.: 1/(1-exp(x)/2). (End)

More terms from Robert G. Wilson v, Oct 29 2002

A002051 Steffensen's bracket function [n,2].

Original entry on oeis.org

0, 0, 1, 9, 67, 525, 4651, 47229, 545707, 7087005, 102247051, 1622631549, 28091565547, 526858344285, 10641342962251, 230283190961469, 5315654681948587, 130370767029070365, 3385534663256714251, 92801587319328148989, 2677687796244383678827, 81124824998504072833245, 2574844419803190382447051

Offset: 1

N. J. A. Sloane

a(n) is the number of ways to arrange the blocks of the partitions of {1,2,...,n} in an undirected cycle of length 3 or more, see A000629. - Geoffrey Critzer, Nov 23 2012
From Gus Wiseman, Jun 24 2020: (Start)
Also the number of (1,2)-matching length-n sequences covering an initial interval of positive integers. For example, the a(2) = 1 and a(3) = 9 sequences are:
  (1,2)  (1,1,2)
         (1,2,1)
         (1,2,2)
         (1,2,3)
         (1,3,2)
         (2,1,2)
         (2,1,3)
         (2,3,1)
         (3,1,2)
Missing from this list are:
  (1,1)  (1,1,1)
  (2,1)  (2,1,1)
         (2,2,1)
         (3,2,1)
(End)

			a(4) = 9. There are 6 partitions of {1,2,3,4} into exactly three blocks and one way to put them in an undirected cycle of length three. There is one partition of {1,2,3,4} into four blocks and 3 ways to make an undirected cycle of length four. 6 + 3 = 9. - _Geoffrey Critzer_, Nov 23 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Steffensen, J. F. Interpolation. 2d ed. Chelsea Publishing Co., New York, N. Y., 1950. ix+248 pp. MR0036799 (12,164d)

T. D. Noe, Table of n, a(n) for n = 1..100
G. J. Simmons, Letter to N. J. Sloane, May 29 1974
J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

A diagonal of the triangular array in A241168.
(1,2)-avoiding patterns are counted by A011782.
(1,1)-matching patterns are counted by A019472.
(1,2)-matching permutations are counted by A033312.
(1,2)-matching compositions are counted by A056823.
(1,2)-matching permutations of prime indices are counted by A335447.
(1,2)-matching compositions are ranked by A335485.
Patterns are counted by A000670 and ranked by A333217.
Patterns matched by compositions are counted by A335456.
Cf. A056986, A335454, A335465, A335486, A335515.

a[n_] := Sum[ k!*StirlingS2[n-1, k], {k, 0, n-1}] - 2^(n-2); Table[a[n], {n, 3, 17}] (* Jean-François Alcover, Nov 18 2011, after Manfred Goebel *)
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],!GreaterEqual@@#&]],{n,0,5}] (* Gus Wiseman, Jun 24 2020 *)

a(n) = sum(s=2, n-1, stirling(n,s+1,2)*s!/2); \\ Michel Marcus, Jun 24 2020

[n,2] = Sum_{s=2..n-1} Stirling2(n,s+1)*s!/2 (cf. A241168).
a(1)=0; for n >= 2, a(n) = A000670(n-1) - 2^(n-2). - Manfred Goebel (mkgoebel(AT)essex.ac.uk), Feb 20 2000; formula adjusted by N. J. A. Sloane, Apr 22 2014. For example, a(5) = 67 = A000670(4)-2^3 = 75-8 = 67.
E.g.f.: (1 - exp(2*x) - 2*log(2 - exp(x)))/4 = B(A(x)) where A(x) = exp(x)-1 and B(x) = (log(1/(1-x))- x - x^2/2)/2. - Geoffrey Critzer, Nov 23 2012

Entry revised by N. J. A. Sloane, Apr 22 2014

A006483 a(n) = Fibonacci(n)*2^n + 1.

Original entry on oeis.org

1, 3, 5, 17, 49, 161, 513, 1665, 5377, 17409, 56321, 182273, 589825, 1908737, 6176769, 19988481, 64684033, 209321985, 677380097, 2192048129, 7093616641, 22955425793, 74285318145, 240392339457, 777925951489, 2517421260801, 8146546327553, 26362777698305

Offset: 0

Dennis S. Kluk (mathemagician(AT)ameritech.net)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (3,2,-4).

Cf. A000045, A063727, A087206.

Equals A103435 + 1.

[Fibonacci(n)*2^n + 1: n in [0..50]]; // Vincenzo Librandi, Jun 09 2013

A006483:=-(-1+6*z**2)/(z-1)/(4*z**2+2*z-1); # Simon Plouffe in his 1992 dissertation

lst={};Do[AppendTo[lst, Fibonacci[n]*2^n+1], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
CoefficientList[Series[(-(- 1 + 6 x^2)) / ((1 - x) (1 - 2 x - 4 x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)
LinearRecurrence[{3,2,-4},{1,3,5},40] (* Harvey P. Dale, Aug 01 2021 *)

G.f.: -(-1+6*x^2)/((1-x)*(1-2*x-4*x^2)).

G.f. in Formula field corrected by Vincenzo Librandi, Jun 09 2013

A053440 Number of k-simplices in the first derived complex of the standard triangulation of an n-simplex. Equivalently, T(n,k) is the number of ascending chains of length k+1 of nonempty subsets of the set {1, 2, ..., n+1}.

Original entry on oeis.org

1, 3, 2, 7, 12, 6, 15, 50, 60, 24, 31, 180, 390, 360, 120, 63, 602, 2100, 3360, 2520, 720, 127, 1932, 10206, 25200, 31920, 20160, 5040, 255, 6050, 46620, 166824, 317520, 332640, 181440, 40320, 511, 18660, 204630, 1020600, 2739240, 4233600, 3780000, 1814400, 362880

Offset: 0

Rob Arthan, Jan 12 2000

T(n,k) is the number of length k+1 sequences of nonempty mutually disjoint subsets of {1,2,...,n+1}. The e.g.f. for the column corresponding to k is exp(x)*(exp(x)-1)^(k+1). - Geoffrey Critzer, Dec 20 2011

			T(2,1) = 12 because there are 12 such length 2 sequences of subsets of {1,2,3}: ({1},{2}), ({1},{3}), ({2},{3}), ({1},{2,3}), ({2},{1,3}), ({3},{1,2}) with two orderings for each. - _Geoffrey Critzer_, Dec 20 2011
Triangle begins:
   1
   3      2
   7     12      6
  15     50     60     24
  31    180    390    360    120

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
F. Brenti and V. Welker, f-vectors of barycentric subdivisions Math. Z., 259(4), 849-865, 2008.
Wikipedia, Barycentric subdivision

Other versions are A028246, A142071.
Columns k=0..1 are A000225(n+1), A028243(n+2).
Cf. A000142 (main diagonal), A002050 (row sums), A019538.

a := (n, k) -> (k+1)!*Stirling2(n+2, k+2):
seq(print(seq(a(n, k), k = 0..n)), n = 0..10);

nn = 5; a = Exp[ x] - 1 ; f[list_] := Select[list, # > 0 &];Map[f, Transpose[Table[Drop[Range[0, nn]!CoefficientList[Series[a^k  Exp[x], {x, 0, nn}],x], 1], {k, 1, 5}]]] // Grid (* Geoffrey Critzer, Dec 20 2011 *)
Table[(k+1)!*StirlingS2[n+2,k+2], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 19 2017 *)

for(n=0,10, for(k=0,n, print1((k+1)!*stirling(n+2,k+2,2), ", "))) \\ G. C. Greubel, Nov 19 2017

T(0,k) = delta(0,k), T(n,k) = delta(0,k) + (k+1)(T(n-1,k-1) + (k+2)T(n-1,k)).
E.g.f.: exp(x)*(exp(x)-1)/(1-y*(exp(x)-1)). - Vladeta Jovovic, Apr 13 2003
T(n,k) = Sum_{i = 0..n} binomial(n+1,i+1)*(k+1)!*Stirling2(i+1,k+1) = (k+1)!*Stirling2(n+2,k+2) (Brenti and Welker). - Peter Bala, Jul 12 2014
T(n,k) = (k+1)!*Stirling2(n+2, k+2). - G. C. Greubel, Nov 19 2017

More terms from James Sellers, Jan 14 2000

A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3x - sqrt(1+6x+x^2)) / (4*x) (A001003).

Original entry on oeis.org

0, 1, -5, 49, -725, 14401, -360005, 10863889, -384415925, 15612336481, -715930020005, 36592369889329, -2062911091119125, 127170577711282561, -8510569547826528005, 614491222512504748369, -47615614242877583230325, 3941408640018910366196641

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..353
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, triassociative operad "Trias".

Cf. A002050, A006531, A084099, A101851, A114285, A225883, A383985, A383986, A383988, A383989, A383991.

Composition of A001003 with exp(x)-1.

nn = 17; f[x_] := (1 + 3*x - Sqrt[1 + 6*x + x^2])/(4*x); Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

Original entry on oeis.org

0, 1, -11, 61, -467, 4381, -49091, 643021, -9615827, 161844541, -3026079971, 62243374381, -1396619164787, 33949401567901, -888725861445251, 24926889744928141, -745755560487363347, 23705772035082494461, -797875590555470224931, 28346366547928396344301

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..407
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, operad "FF6".

Cf. A002050, A006531, A084099, A101851, A114285, A225883, A383985, A383986, A383987, A383988, A383995.

nn = 19; f[x_] := x*(1 - 3*x - x^2 + x^3)/(1 + 3*x + x^2 - x^3);
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A383993 Series expansion of the exponential generating function exp(tridup^!(x)) - 1 where tridup^!(x) = x / ((1+x) * (1+2*x)).

Original entry on oeis.org

0, 1, -5, 25, -119, 301, 5611, -171275, 3574705, -68597639, 1282415131, -23479249199, 409082338105, -6146707844315, 46462772999371, 2072826643602541, -160983324879816479, 8004468391727017585, -352443295329194182085, 14817357881274444545161

Offset: 0

Michael De Vlieger, May 16 2025

The series tridup^!(x) is the inverse for the substitution of the series tridup(x) (given by A001003), given by the suspension of the Koszul dual of tridup. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..404
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, triduplicial operad "TriDup".

Cf. A002050, A003725, A097388, A111884, A112242, A177885, A318215, A383990, A383991, A383992, A383994, A383995.

nn = 19; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[x/((1 + x)*(1 + 2*x))], {x, 0, nn}], x]

A006488 Numbers n such that n! has a square number of digits.

Original entry on oeis.org

0, 1, 2, 3, 7, 12, 18, 32, 59, 81, 105, 132, 228, 265, 284, 304, 367, 389, 435, 483, 508, 697, 726, 944, 1011, 1045, 1080, 1115, 1187, 1454, 1494, 1617, 1659, 1788, 1921, 2012, 2105, 2200, 2248, 2395, 2445, 2861, 2915, 3192, 3480, 3539, 3902, 3964, 4476

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Numbers whose square is represented by the number of digits of n!: 1, 2, 3, 4, 6, 9, 11, 13, 15, 21, 23, 24, 25, 28, 29, ..., . - Robert G. Wilson v, May 14 2014
From Bernard Schott, Jan 04 2020: (Start)
In M. Gardner's book, see reference, there is a tree printout of 105! with 169 digits, where the bottom row consists of the 25 trailing zeros of 105!. M. Gardner does not explain if this is the only factorial that can be displayed in a similar tree form.
Proof: If m! has q^2 digits, hence the number of trailing zeros in m! must be equal to 2*q-1 to satisfy this triangular look; m = 105 satisfies these two conditions with q = 13 because 105! has 13^2 = 169 digits and 2*13-1 = 25 trailing zeros.
When m < 105 and m! has q^2 digits (m <= 81), then q <= 11 and the number of trailing zeros is <= 2*q - 3.
When m > 105 and m! has q^2 digits (m >= 132), then q >= 15 and the number of trailing zeros is >= 2*q + 2.
Hence, only 105! presents such a tree printout.
              1
             081
            39675
           8240290
          900504101
         30580032964
        9720646107774
       902579144176636
      57322653190990515
     3326984536526808240
    339776398934872029657
   99387290781343681609728
  0000000000000000000000000
(End)

M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 55.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..1311
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979
Eric Weisstein's World of Mathematics, Stirling's Approximation and Stirling's Series
Index entries for sequences related to factorial numbers

Cf. A000142, A027868 (trailing zeros), A034886 (number of digits), A056851.

[k:k in [0..5000]| IsSquare(#Intseq(Factorial(k)))]; // Marius A. Burtea, Jan 04 2020

LogBase10Stirling[n_] := Floor[Log[10, 2 Pi n]/2 + n*Log[10, n/E] + Log[10, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5)]]; Select[ Range[ 4500], IntegerQ[ Sqrt[ (LogBase10Stirling[ # ] + 1)]] & ] (* The Mathematica coding comes from J. Stirling's expansion for the Gamma function; see the links. For more terms inside the last Log_10 function, use A001163 & A001164. Robert G. Wilson v, Apr 27 2014 *)
Select[Range[0,4500],IntegerQ[Sqrt[IntegerLength[#!]]]&] (* Harvey P. Dale, Sep 27 2018 *)

isok(n) = issquare(#Str(n!)); \\ Michel Marcus, Sep 05 2015

A241168 Triangle read by rows: T(n,k) (1 <= k <= n) = Steffensen's bracket function [n,n-k].

Original entry on oeis.org

1, 1, 2, 1, 5, 6, 1, 9, 25, 26, 1, 14, 67, 149, 150, 1, 20, 145, 525, 1081, 1082, 1, 27, 275, 1450, 4651, 9365, 9366, 1, 35, 476, 3430, 15421, 47229, 94585, 94586, 1, 44, 770, 7266, 43281, 180894, 545707, 1091669, 1091670, 1, 54, 1182, 14154, 107751, 581280, 2359225, 7087005, 14174521, 14174522

Offset: 1

N. J. A. Sloane, Apr 22 2014

Steffensen's bracket function [n,k] = Sum_{s=k..n-1} Stirling2(n,s+1)*s!/k!.

The numbers are used in numerical integration.

			Triangle begins:
1,
1, 2,
1, 5, 6,
1, 9, 25, 26,
1, 14, 67, 149, 150,
1, 20, 145, 525, 1081, 1082,
1, 27, 275, 1450, 4651, 9365, 9366,
1, 35, 476, 3430, 15421, 47229, 94585, 94586,
1, 44, 770, 7266, 43281, 180894, 545707, 1091669, 1091670,
...

J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, 11 (1928), 75-97.

J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.

Diagonals include A000096, A000629, A002050, A002051, A241169, A241170.

with(combinat);
T:=proc(n,k) add(stirling2(n,s+1)*s!/k!,s=k..n-1); end;
for n from 1 to 12 do lprint([seq(T(n,n-k),k=1..n)]); od:

cp's OEIS Frontend

A052856 E.g.f.: (1-3*exp(x)+exp(2*x))/(exp(x)-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A076726 a(n) = Sum_{k>=0} k^n/2^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A002051 Steffensen's bracket function [n,2].

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A006483 a(n) = Fibonacci(n)*2^n + 1.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A053440 Number of k-simplices in the first derived complex of the standard triangulation of an n-simplex. Equivalently, T(n,k) is the number of ascending chains of length k+1 of nonempty subsets of the set {1, 2, ..., n+1}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3*x - sqrt(1+6*x+x^2)) / (4*x) (A001003).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3*x-x^2+x^3) / (1+3*x+x^2-x^3).

Views

Author

Keywords

Links

A052856 E.g.f.: (1-3exp(x)+exp(2x))/(exp(x)-2).

A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3x - sqrt(1+6x+x^2)) / (4*x) (A001003).

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).