A184011 -id:A184011 - cp's OEIS frontend

A309509 G.f. satisfies A(A(x)) = F(x), where F(x) is the g.f. for A001787(n) = n*2^(n-1).

0, 1, 2, 2, 2, 2, 0, 4, 6, -58, 100, 1052, -5924, -21972, 322020, 332392, -21168682, 29068598, 1724404180, -7070346036, -172304798980, 1290100381724, 20728501384592, -247269172883976, -2936888518668676, 53037176259027580, 477640220538178184

Offset: 0

Vladimir Reshetnikov, Aug 05 2019

sign

A(x) is sometimes called a functional square root, or half-iterate of F(x).

Seiichi Manyama, Table of n, a(n) for n = 0..488
Gottfried Helms, Coefficients for fractional iterates exp(x)-1.
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986 [math.CO], 2013.
Wikipedia, Functional square root.

Cf. A001787, A027436, A030274, A030275, A184011.

half[q_] := half[q] = Module[{h}, h[0] = 0; h[1] = 1; h[n_Integer] := h[n] = Module[{c}, c[m_Integer /; m < n] := h[m]; c[n] /. Solve[q[n] == Sum[k! c[k] BellY[n, k, Table[m! c[m], {m, n - k + 1}]], {k, n}]/n!, c[n]][[1]]]; h]; a[n_Integer] := a[n] = half[Function[k, k 2^(k-1)]][n]; Table[a[n], {n, 0, 26}]

Define the sequence b(n,k) as follows. If nSeiichi Manyama, May 03 2024

A209519 Expansion A(x) = Sum_{n>0} a(n)x^n/(3^(n-1)n!), A(x) satisfies A(A(A(x)))=e^x-1.

Original entry on oeis.org

1, 1, 0, 0, 2, -21, 138, 150, -22833, 303975, 3451320, -214016553, 666006714, 228865308144, -4943013567642, -396567325158381, 21423378444873687, 1022158819761317838, -121532275123709160942

Offset: 1

Vladimir Kruchinin, Mar 10 2012

sign

Seiichi Manyama, Table of n, a(n) for n = 1..200

Cf. A184011.

T(n,m):=if n=m then 1 else 1/3*(stirling2(n,m)*m!/n!-sum(T(k,m)*sum(T(n,i)*T(i,k),i,k,n),k,m+1,n-1)-T(m,m)*sum(T(n,i)*T(i,m),i,m+1,n-1));
makelist(n!*3^(n-1)*(T(n,1)),n,1,7);

a(n)=3^(n-1)*n!*T(n,1), T(n,m)=1/3*(stirling2(n,m)*m!/n!-sum(k=m+1..n-1, T(k,m)*sum(i=k..n, T(n,i)*T(i,k)))-T(m,m)*sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1.

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

A199203 Decimal expansion of f(0) where f is the functional square root (half-iterate) of exponent, f(f(x))=exp(x).

Original entry on oeis.org

4, 9, 8, 5, 6, 3

Offset: 0

Vladimir Reshetnikov, Nov 03 2011

f(A199203) = 1, where f is the functional square root of exponent.

The listed digits are a conjecture obtained as a common result of several different numeric approximation algorithms, they haven't been rigorously proved to be correct.

			0.498563...

Citizendium, Superfunction.
Math StackExchange, Half iteration of exponential function.
Gottfried Helms, Coefficients for fractional iterates exp(x)-1.
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.
Wikipedia, Functional square root.

Cf. A001113, A184011.

n = 23; p[s_, 1] := c[s]; p[0, n_] := c[0]^n; p[s_, n_] := p[s, n] = Expand[Sum[c[k] p[s - k, n - 1], {k, 0, s}]]; d[n_, 0] := Sum[c[k] c[0]^k, {k, 0, n}]; d[n_, i_] := Sum[c[k] p[i, k], {k, 1, n}]; a = c[0] /. FindRoot[Table[d[n, k] k! == 1, {k, 0, n}], Table[{c[k], 2^-k}, {k, 0, n}], WorkingPrecision -> 30]; First[RealDigits[a, 10, 6]]

A372746 E.g.f. A(x) satisfies A(A(A(A(A(x))))) = (exp(5*x) - 1)/5.

Original entry on oeis.org

0, 1, 1, -1, 3, -5, -140, 3265, -35145, -423300, 29156450, -244674300, -35711239900, 1323533263450, 79091925545150, -7803803198306500, -249393437031632750, 68360119098495114000, 386044168785212367500, -899401682046014451577000

Offset: 0

Seiichi Manyama, May 12 2024

sign

Cf. A184011, A209519.

Define the sequence b(n,m) as follows. If n

A372795 E.g.f. A(x) satisfies A(A(A(A(x)))) = (exp(8*x) - 1)/8.

Original entry on oeis.org

0, 1, 2, -2, 8, 24, -2240, 59600, -640000, -35477120, 2287843200, -337824000, -8328489693696, 320219485774848, 53149588906171392, -5832590252624818176, -534898113615540043776, 142559169839206640025600, 6582786304965587026329600

Offset: 0

Seiichi Manyama, May 13 2024

sign

Cf. A184011, A209519, A372746.

Cf. A372796.

Define the sequence b(n,m) as follows. If n

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)!)).

Showing 1-8 of 8 results.

cp's OEIS Frontend

A309509 G.f. satisfies A(A(x)) = F(x), where F(x) is the g.f. for A001787(n) = n*2^(n-1).

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A209519 Expansion A(x) = Sum_{n>0} a(n)*x^n/(3^(n-1)*n!), A(x) satisfies A(A(A(x)))=e^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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A199203 Decimal expansion of f(0) where f is the functional square root (half-iterate) of exponent, f(f(x))=exp(x).

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A372746 E.g.f. A(x) satisfies A(A(A(A(A(x))))) = (exp(5*x) - 1)/5.

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A372795 E.g.f. A(x) satisfies A(A(A(A(x)))) = (exp(8*x) - 1)/8.

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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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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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A209519 Expansion A(x) = Sum_{n>0} a(n)x^n/(3^(n-1)n!), A(x) satisfies A(A(A(x)))=e^x-1.

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.