cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

N. J. A. Sloane, Jan 23 2000

Keywords

Examples

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

References

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

Links

Crossrefs

Cf. A052123, A052104, A052105.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

More terms from Vladeta Jovovic, Jul 27 2002

A052123 Log_2 of denominators of coefficients in the e.g.f. a(x) such that a(a(x)) = exp(x) - 1.

Original entry on oeis.org

0, 0, 1, 3, 0, 5, 7, 7, 8, 8, 12, 14, 14, 15, 16, 18, 18, 19, 21, 24, 24, 26, 28, 29, 26, 27, 32, 34, 35, 35, 37, 36, 38, 39, 40, 41, 43, 46, 48, 49, 47, 50, 53, 55, 55, 56, 58, 59, 57, 56, 61, 64, 64, 66, 68, 70, 70, 71, 73, 75, 76, 76, 76, 78, 78, 79, 80, 82
Offset: 0

Views

Author

N. J. A. Sloane, Jan 23 2000

Keywords

Examples

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

References

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

Crossrefs

Cf. A052122, A052104, A052105.

Programs

  • Mathematica
    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] // Denominator // Log[2, #]&, {n, 0, 29}] (* Jean-François Alcover, Mar 03 2014, after A052104 and Alois P. Heinz *)

Extensions

More terms from Vladeta Jovovic, Jul 27 2002
More terms from Sean A. Irvine, Oct 25 2021

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

Views

Author

N. J. A. Sloane, Jan 22 2000

Keywords

Examples

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

References

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

Links

Crossrefs

Cf. A052105, A052122, A052123.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • Sage
    @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

Formula

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

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

Views

Author

Thomas Scheuerle, Mar 10 2025

Keywords

Comments

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)!.

Examples

			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)
+ ... .

Links

Crossrefs

Cf. A381932 (numerators).
Cf. A052123, A052122, A052104, A052105, A144150, A180609, A184011.

Programs

  • PARI
    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)));

Formula

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

Views

Author

Thomas Scheuerle, Mar 12 2025

Keywords

Comments

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

Examples

			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)
+ ... .

Crossrefs

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

Programs

  • PARI
    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)));

Formula

Conjecture: abs(T(n, 2)) = A064169(n - 1).
T(n, n) = numerator(A180609(n)/(n!*(n+1)!)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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