A180609 -id:A180609 - cp's OEIS frontend

A134242 Numerators of certain constants c_n = A180609(n)/n! related to Hurwitz numbers.

1, -1, 1, -2, 11, -3, -11, 29, 493, -2711, -12406, 2636317, -10597579, -439018457, 1165403153, 118734633647, -105428488301, -4070802683898, 1695077946695371, 56532812889378221, -252968859037883917, -425882179787933647571, 123624959518930226565553, 32729394708071881944913, -5814212300444136523052695

Offset: 1

N. J. A. Sloane, Jan 30 2008

Manetti-Ricciardi refer to the c_n as Koszul numbers.

			The fractions are 1, -1/2, 1/2, -2/3, 11/12, -3/4, -11/6, 29/4, 493/12, -2711/6, -12406/15,  2636317/60, -10597579/120, -439018457/60, 1165403153/20, 118734633647/60, ...

M Manetti, G Ricciardi, Universal Lie formulas for higher antibrackets, arXiv preprint arXiv:1509.09032 [math.QA], 2015-2016.
S. Shadrin and D. Zvonkine, Changes of variables in ELSV-type formulas, Michigan Mathematical Journal, vol. 55 (2007), 209-228.
D. Zvonkine, Home Page

Cf. A134243, A180609.

K[1] = 1;
K[n_] := K[n] = -2/((n+2)(n-1)) Sum[StirlingS2[n+1, i] K[i], {i, 1, n-1}];
Table[Numerator[K[n]], {n, 1, 25}] (* Jean-François Alcover, Jul 26 2018 *)

Manetti-Ricciardi Theorem 4.4 give a recurrence for the c_n in terms of Stirling numbers.

More terms from Manetti-Ricciardi added by N. J. A. Sloane, May 25 2016

A134243 Denominators of certain constants c_n = A180609(n)/n! related to Hurwitz numbers.

Original entry on oeis.org

1, 2, 2, 3, 12, 4, 6, 4, 12, 6, 15, 60, 120, 60, 20, 60, 3, 5, 60, 120, 8, 1260, 2520, 168, 56, 168, 168, 840, 84, 840, 21, 140, 420, 630, 120, 280, 420, 840, 504, 2520, 840, 840, 315, 2520, 2520, 315, 84, 90, 30, 180, 360, 120, 120, 210, 24, 495, 1980, 2640, 55440, 315, 55440, 45, 2772, 6930, 27720, 9240, 770, 1848, 27720, 27720

Offset: 1

N. J. A. Sloane, Jan 30 2008

Manetti-Ricciardi refer to the c_n as Koszul numbers.

			The fractions are 1, -1/2, 1/2, -2/3, 11/12, -3/4, -11/6, 29/4, 493/12, -2711/6, -12406/15, 2636317/60, -10597579/120, -439018457/60, 1165403153/20, 118734633647/60, ...

Marco Manetti and Giulia Ricciardi, Universal Lie formulas for higher antibrackets, arXiv preprint arXiv:1509.09032 [math.QA], 2015-2016.
Sergey Shadrin and Dimitri Zvonkine, Changes of variables in ELSV-type formulas, Michigan Mathematical Journal, vol. 55 (2007), 209-228.
Dimitri Zvonkine, Home Page

Cf. A134242, A180609.

K[1] = 1;
K[n_] := K[n] = -2/((n+2)(n-1)) Sum[StirlingS2[n+1, i] K[i], {i, 1, n-1}];
Table[Denominator[K[n]], {n, 1, 70}] (* Jean-François Alcover, Jul 26 2018 *)

Manetti-Ricciardi Theorem 4.4 give a recurrence for the c_n in terms of Stirling numbers.

More terms from Manetti-Ricciardi added by N. J. A. Sloane, May 25 2016

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

A134242 Numerators of certain constants c_n = A180609(n)/n! related to Hurwitz numbers.

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A134243 Denominators of certain constants c_n = A180609(n)/n! related to Hurwitz numbers.

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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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Examples

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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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

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.