A052123 -id:A052123 - cp's OEIS frontend

A184011 Coefficients of the formal power series of a half-iterate of exp(x)-1 (rescaled).

0, 1, 2, 2, 0, 8, -56, 32, 10176, -215808, -78784, 150990912, -3405688576, -139041794560, 10385778676736, 130003936220160, -43016304236761088, 526545841919713280, 266085261164348628992, -12347306589339686547456

Offset: 0

Gottfried Helms, Feb 13 2011

sign

Consider the formal power series for the real half-iterate of exp(x)-1 = Sum_{k>=0} c_k*x^k with c_1 = +1 then a(k) = c_k*k!*4^{k-1} and all a(k) seem to be integers.

For the general technique of finding the half-iterate of power series, see for instance the Comtet reference.

			f(x) = x + 1/4*x^2 + 1/48*x^3 + 1/3840*x^5 - 7/92160*x^6 + 1/645120*x^7 + O(x^8)
so   c_3  = 1/48
and  a(3) = c_3 * 4^2*3! = 16*6/48 = 2

Comtet, L; Advanced Combinatorics (1974 edition), D. Reidel Publishing Company, Dordrecht - Holland, pp. 147-148.

Gottfried Helms, Coefficients for fractional iterates exp(x)-1
Dmitry Kruchinin and Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986 [math.CO], 2013.

Cf. A052122, A052123.

max = 19; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = 1; coes = CoefficientList[ Series[f[f[x]] - Exp[x] - 1, {x, 0, max}], x]; sol = Solve[Thread[coes == 0] // Rest] // First; Table[c[n]*4^(n-1)*n!, {n, 0, max}] /. sol (* Jean-François Alcover, Feb 11 2013 *)

{a(n)=local(A=x+x^2,B=x);for(i=1,n,B=serreverse(A+x*O(x^n));A=(A+exp(B)-1)/2);4^(n-1)*n!*polcoeff(A,n)} \\ Paul D. Hanna

{trisqrt(m) = local(tmp, rs=rows(m), cs=cols(m), c);
\\ computes sqrt of lower triangular matrix with unit-diagonal
   tmp=matid(#m);
   for(d=1,rs-1,
        for(r=d+1,rs,
              c=r-d;
              tmp[r,c]=(m[r,c]-sum(k=c+1,r-1,tmp[r,k]*tmp[k,c]))
                        /(tmp[c,c]+tmp[r,r])
           );
      );
return(tmp);}
ff = exp(x)-1
Mff = matrix(6,6,r,c,polcoeff(ff^(c-1),(r-1))) \\ create Bell-matrix for ff
Mf =  trisqrt ( Mff )  \\ = Mff^(1/2) is Bellmatrix for f
f = Ser(Mf[,2])  \\ coefficients of power series for half-iterate of exp(x)-1 from second column in Mf

G.f. f(x) where f(f(x)) = exp(x)-1 with f'(0)=1.
T(n,m) = if n=m then 1 else (stirling2(n,m)*m!/n!-sum(i=m+1..n-1, T(n,i)*T(i,m)))/2; a(n) = 4^(n-1)*n!*T(n,1). - Vladimir Kruchinin, Nov 09 2011
E.g.f. A(x), satisfies A(A(x))=(exp(4*x)-1)/4, T(n,m)=1/2*(4^(n-m)*stirling2(n,m)-sum(i=m+1..n-1, T(n,i)*T(i,m))), T(n,n)=1, a(n)=T(n,1), a(0)=0. - Dmitry Kruchinin, Dec 04 2012
a(n) = A052122(n) * 2^(2*n - 2 - A052123(n)). - Andrey Zabolotskiy, Aug 22 2022

A052122 Numerators of coefficients in the e.g.f. a(x) such that a(a(x)) = exp(x) - 1.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, -7, 1, 159, -843, -1231, 2359233, -13303471, -271566005, 10142361989, 126956968965, -10502027401553, 64275615468715, 32481110981976151, -3014479147788009411, -147131182752475409229, 14607119841651449406947, 1868869263315549659372569

Offset: 0

N. J. A. Sloane, Jan 23 2000

			a(x) = x + 1/4*x^2 + 1/48*x^3 + 1/3840*x^5 - 7/92160*x^6 + 1/645120*x^7 + ...

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.52.

Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986 [math.CO], 2013.

Cf. A052123, A052104, A052105.

T[n_, n_] = 1; T[n_, m_] := T[n, m] = (StirlingS2[n, m]*m!/n! - Sum[T[n, i]*T[i, m], {i, m+1, n-1}])/2; Table[n!*T[n, 1] // Numerator , {n, 0, 22}] (* Jean-François Alcover, Mar 03 2014, after A052104 and Alois P. Heinz *)

a(n)/2^A052123(n) = n!*A052104(n)/A052105(n). - R. J. Mathar, Sep 25 2011

More terms from Vladeta Jovovic, Jul 27 2002

A052104 Numerators of coefficients of the formal power series a(x) such that a(a(x)) = exp(x) - 1.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, -7, 1, 53, -281, -1231, 87379, -13303471, -54313201, 10142361989, 2821265977, -10502027401553, 1836446156249, 2952828271088741, -1004826382596003137, -7006246797736924249, 14607119841651449406947, 1868869263315549659372569

Offset: 0

N. J. A. Sloane, Jan 22 2000

			a(x) = x + x^2/4 + x^3/48 + x^5/3840 - 7*x^6/92160 + x^7/645120 + ...

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.52c.

Alois P. Heinz, Table of n, a(n) for n = 0..120
I. N. Baker, Zusammensetzungen ganzer Funktionen Math. Z. 69 (1) (1958) 121-163.
Dmitry Kruchinin and Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.
Mathoverflow, f(f(x))=exp(x)-1 and...

Cf. A052105, A052122, A052123.

T:= proc(n, m) T(n, m):= `if`(n=m, 1, (Stirling2(n, m)*m!/n!-
       add(T(n,i)*T(i,m), i=m+1..n-1))/2)
    end:
a:= n-> numer(T(n, 1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 11 2013

T[n_, n_] = 1; T[n_, m_] := T[n, m] = (StirlingS2[n, m]*m!/n! - Sum[T[n, i]*T[i, m], {i, m+1, n-1}])/2; Table[T[n, 1] // Numerator, {n, 0, 30}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)

@CachedFunction
def T(n,k):
    if (k==n): return 1
    else: return ( (factorial(k)/factorial(n))*stirling_number2(n,k) - sum(T(n,j)*T(j,k) for j in (k+1..n-1)) )/2
[numerator(T(n,1)) for n in (0..30)] # G. C. Greubel, Apr 15 2021

a(n) = numerator(T(n,1)) where T(n, m) = if n=m then 1, otherwise ( StirlingS2(n, m)*m!/n! - Sum_{i=m+1..n-1} T(n, i)*T(i, m) )/2. - Vladimir Kruchinin, Nov 08 2011

A052105 Denominators of coefficients in the formal power series a(x) such that a(a(x)) = exp(x) - 1.

Original entry on oeis.org

1, 1, 4, 48, 1, 3840, 92160, 645120, 3440640, 30965760, 14863564800, 24222105600, 7847962214400, 40809403514880, 5713316492083200, 7617755322777600, 5484783832399872000, 5328075722902732800, 1220613711064989696000

Offset: 0

N. J. A. Sloane, Jan 22 2000

			a(x) = x + x^2/4 + x^3/48 + x^5/3840 - 7*x^6/92160 + x^7/645120 + ...

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.52c.

Alois P. Heinz, Table of n, a(n) for n = 0..120
Dmitry Kruchinin and Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.

Cf. A052104, A052122, A052123.

T:= proc(n, k);
T(n, k):= `if`(n=k, 1, (Stirling2(n, k)*k!/n! - add(T(n, j)*T(j, k), j = k+1..n-1))/2);
end proc;
a:= n -> denom(T(n, 1));
seq(a(n), n = 0..30); # G. C. Greubel, Apr 15 2021

(* First program *)
a[x_, n_] := Sum[c[k] x^k, {k, 0, n}] ;
f[x_, n_] := Series[Exp[x] - 1, {x, 0, n}] // Normal;
b[x_, n_] := Series[a[a[x, n], n], {x, 0, n}] // Normal;
eq[n_] := Thread[CoefficientList[f[x, n] - b[x, n], x] == 0] // Rest;
c[0] = 0; so[3] = Solve[eq[3], {c[1], c[2], c[3]}] // First;
so[n_] := so[n] = Solve[eq[n] /. Flatten[Table[so[k], {k, 3, n - 1}]], c[n]] // First
Array[c, 19, 0] /. Flatten[Table[so[k], {k, 3, 19}]] // Denominator
(* Jean-François Alcover, Jun 08 2011 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==n, 1, (StirlingS2[n, k]*k!/n! - Sum[T[n, j]*T[j, k], {j, k+1, n-1}])/2];
Table[Denominator[T[n, 1]], {n, 0, 30}] (* G. C. Greubel, Apr 15 2021 *)

@CachedFunction
def T(n,k):
    if (k==n): return 1
    else: return ( (factorial(k)/factorial(n))*stirling_number2(n,k) - sum(T(n,j)*T(j,k) for j in (k+1..n-1)) )/2
[denominator(T(n,1)) for n in (0..30)] # G. C. Greubel, Apr 15 2021

a(n) = denominator(T(n,1)) where T(n, m) = if n=m then 1, otherwise ( StirlingS2(n, m)*m!/n! - Sum_{i=m+1..n-1} T(n, i)*T(i, m) )/2. - G. C. Greubel, Apr 15 2021

A220112 E.g.f. A(x) satisfies A(A(x)) = (1/4)log(1/(1-4x)).

Original entry on oeis.org

1, 2, 10, 80, 872, 11928, 195072, 3702080, 80065792, 1950808000, 53016791360, 1587229842688, 51619520360960, 1808576831681536, 68562454975587328, 2830905156661645312, 124395772159835529216, 5504660984739184156672, 250011277837808237105152, 14799530615476409472303104

Offset: 1

Dmitry Kruchinin, Dec 05 2012

sign

a(23) = -4050933314339181211663673622528 is the first negative term. - Vladimir Reshetnikov, Aug 15 2021

Comtet, L; Advanced Combinatorics (1974 edition), D. Reidel Publishing Company, Dordrecht - Holland, pp. 147-148.

Vladimir Reshetnikov, Table of n, a(n) for n = 1..281
Gottfried Helms, Coefficients for fractional iterates exp(x)-1
Dmitry Kruchinin and Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986 [math.CO], 2013

Cf. A052122, A052123, A184011.

A := proc(n, m) option remember; if n = m then 1 else
1/2*(4^(n-m)*(-1)^(n-m)*Stirling1(n,m) - add(A(n,k)*A(k,m), k =m+1..n-1)) fi end: a := n -> A(n,1): seq(a(n), n = 1..23); # Peter Luschny, Aug 15 2021

t[n_, m_] := t[n, m] = 1/2*(4^(n - m)*(-1)^(n - m)*StirlingS1[n, m] - Sum[t[n, i]*t[i, m], {i, m+1, n-1}]); t[n_, n_] = 1; Table[t[n, 1], {n, 1, 20}] (* Jean-François Alcover, Feb 22 2013 *)

T(n,m):=if n=m then 1 else 1/2*(4^(n-m)*(-1)^(n-m)*stirling1(n,m)-sum(T(n,i)*T(i,m),i,m+1,n-1));
makelist((T(n,1)),n,1,10);

a(n) = T(n,1), T(n,m) = (1/2)*(4^(n-m)*(-1)^(n-m)*Stirling1(n,m) - Sum_{i=m+1..n-1} T(n,i)*T(i,m)), T(n,n)=1.

More terms from Vladimir Reshetnikov, Aug 15 2021

A381931 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)Sum_{k=1..n} r^(n+1-k)A381932(n, k)/T(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

Original entry on oeis.org

2, 4, 12, 8, 48, 48, 16, 144, 24, 180, 32, 1152, 1728, 5760, 8640, 64, 640, 3456, 5760, 17280, 6720, 128, 7680, 34560, 1152, 34560, 32256, 241920, 256, 26880, 82944, 414720, 41472, 580608, 107520, 1451520, 512, 430080, 645120, 622080, 4147200, 6967296, 21772800, 87091200, 43545600

Offset: 1

Thomas Scheuerle, Mar 10 2025

This is the main entry for this sequence of fractions.
Convergence and analytic continuation of this series representation are interesting research topics with many unsolved problems and open questions.
Evaluating the polynomial of row n P(x) = Sum_{k=1..n} x^(n+1-k)*A381932(n, k)/T(n, k) gives A144150(n+1, x-1)/(n+1)!.

			Triangle T(n, k) begins:
[1]  2;
[2]  4,   12;
[3]  8,   48,     48;
[4]  16,  144,    24,     180;
[5]  32,  1152,   1728,   5760,   8640;
[6]  64,  640,    3456,   5760,   17280,   6720;
[7]  128, 7680,   34560,  1152,   34560,   32256,   241920;
[8]  256, 26880,  82944,  414720, 41472,   580608,  107520,   1451520;
[9]  512, 430080, 645120, 622080, 4147200, 6967296, 21772800, 87091200, 43545600;
.
F^{r}(x) = x
+ x^2*1/2*r
+ x^3*(1/4*r^2 - 1/12*r)
+ x^4*(1/8*r^3 - 5/48*r^2 + 1/48*r)
+ x^5*(1/16*r^4 - 13/144*r^3 + 1/24*r^2 - 1/180*r)
+ x^6*(1/32*r^5 - 77/1152*r^4 + 89/1728*r^3 - 91/5760*r^2 + 11/8640*r)
+ ... .

Gottfried Helms, Coefficients for fractional iterates exp(x)-1
MathOverflow, f(f(x)) = exp(x)-1 and other functions "just in the middle" between linear and exponential

Cf. A381932 (numerators).

Cf. A052123, A052122, A052104, A052105, A144150, A180609, A184011.

c(k, n) = {my(f=x); for(m=1, k, f=subst(f, x, exp(x)-1)); polcoeff(f+O(x^(n+1)), n)}
row(n) = my(p=polinterpolate(vector(2*(n+1), k, k-1), vector(2*(n+1), k, c(k-1, n+1)))); vector(n, k, denominator(polcoeff(p, n-k+1)));

T(n, 1) = 2^n.

T(n, n) = denominator(A180609(n)/(n!*(n+1)!)).

A381932 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)Sum_{k=1..n} r^(n+1-k)T(n, k)/A381931(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

Original entry on oeis.org

1, 1, -1, 1, -5, 1, 1, -13, 1, -1, 1, -77, 89, -91, 11, 1, -29, 175, -149, 91, -1, 1, -223, 1501, -37, 391, -43, -11, 1, -481, 2821, -13943, 725, -2357, 17, 29, 1, -4609, 16099, -19481, 91313, -55649, 23137, 1727, 493, 1, -4861, 89993, -933293, 399637, -1061231, 2035739, -8189, 4897, -2711

Offset: 1

Thomas Scheuerle, Mar 12 2025

The main entry for this sequence of fractions is in A381931.

			Triangle T(n, k) begins:
[1]  1;
[2]  1,    -1;
[3]  1,    -5,     1;
[4]  1,   -13,     1,     -1;
[5]  1,   -77,    89,    -91,    11;
[6]  1,   -29,   175,   -149,    91,     -1;
[7]  1,  -223,  1501,    -37,   391,    -43,   -11;
[8]  1,  -481,  2821, -13943,   725,  -2357,    17,   29;
[9]  1, -4609, 16099, -19481, 91313, -55649, 23137, 1727, 493;
.
F^{r}(x) = x
+ x^2*1/2*r
+ x^3*(1/4*r^2 - 1/12*r)
+ x^4*(1/8*r^3 - 5/48*r^2 + 1/48*r)
+ x^5*(1/16*r^4 - 13/144*r^3 + 1/24*r^2 - 1/180*r)
+ x^6*(1/32*r^5 - 77/1152*r^4 + 89/1728*r^3 - 91/5760*r^2 + 11/8640*r)
+ ... .

Cf. A381931 (denominators).
Cf. A052123, A052122, A052104, A052105
Cf. A064169, A144150, A180609, A184011.

c(k, n) = {my(f=x); for(m=1, k, f=subst(f, x, exp(x)-1)); polcoeff(f+O(x^(n+1)), n)}
row(n) = my(p=polinterpolate(vector(2*(n+1), k, k-1), vector(2*(n+1), k, c(k-1, n+1)))); vector(n, k, numerator(polcoeff(p, n-k+1)));

Conjecture: abs(T(n, 2)) = A064169(n - 1).

T(n, n) = numerator(A180609(n)/(n!*(n+1)!)).

cp's OEIS Frontend

A184011 Coefficients of the formal power series of a half-iterate of exp(x)-1 (rescaled).

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A052122 Numerators of coefficients in the e.g.f. a(x) such that a(a(x)) = exp(x) - 1.

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A052104 Numerators of coefficients of the formal power series a(x) such that a(a(x)) = exp(x) - 1.

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A052105 Denominators of coefficients in the formal power series a(x) such that a(a(x)) = exp(x) - 1.

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A220112 E.g.f. A(x) satisfies A(A(x)) = (1/4)*log(1/(1-4*x)).

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A381931 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)*Sum_{k=1..n} r^(n+1-k)*A381932(n, k)/T(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

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Author

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Comments

Examples

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Programs

PARI

Formula

A381932 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)*Sum_{k=1..n} r^(n+1-k)*T(n, k)/A381931(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

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A220112 E.g.f. A(x) satisfies A(A(x)) = (1/4)log(1/(1-4x)).

A381931 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)Sum_{k=1..n} r^(n+1-k)A381932(n, k)/T(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.

A381932 Triangular array T(n, k) read by rows: denominators of the coefficients for the iterated exponential F^{r}(x) = x + Sum_{n>=1} x^(n+1)Sum_{k=1..n} r^(n+1-k)T(n, k)/A381931(n, k) with F^{1}(x) = exp(x)-1 and F^{2}(x) = exp(exp(x)-1)-1.