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 1-3 of 3 results.

A132094 Numerators of expansion of e.g.f. x^2/(2*(cos(x)-1)), even powers only.

Original entry on oeis.org

-1, -1, -1, -5, -7, -15, -7601, -91, -3617, -745739, -3317609, -5981591, -5436374093, -213827575, -213745149261, -249859397004145, -238988952277727, -28354566442037, -26315271553053477373, -108409774812137683, -3394075340453838586663, -62324003400640902910331
Offset: 1

Views

Author

Jonathan Vos Post, Aug 09 2007

Keywords

Comments

Numerators and denominators given only for even n (odd n have numerators = 0).

Examples

			-1, 0, -1/6, 0, -1/10, 0, -5/42, 0, -7/30, 0, -15/22, 0, -7601/2730, 0.

References

  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.

Links

Crossrefs

Denominators are A132095.
Cf. A132092-A132099.

Programs

  • Maple
    A132094 := proc(n) add( 2*(-1)^i*x^(2*i)/(2*i+2)!,i=0..n+1) ; numer(coeftayl(-1/%,x=0,n)*n!) ; end: for n from 0 to 46 by 2 do printf("%d, ",A132094(n)) ; od: # R. J. Mathar, Oct 18 2007
  • Mathematica
    A132094[n_] := (s = Sum[ 2*(-1)^i*x^(2*i)/(2*i + 2)!, {i, 0, n + 1}]; Numerator[SeriesCoefficient[-1/s, {x, 0, n}]*n!]);
Table[A132094[n], {n, 0, 46, 2}] (* Jean-François Alcover, Nov 24 2017, after R. J. Mathar *)
  • PARI
    my(x='x+O('x^50), v=apply(numerator, Vec(serlaplace(x^2/(2*(cos(x)-1)))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Jan 25 2024

Formula

Asymptotic series 2*Psi(1,x) + x*Psi(2,x) ~ Sum_{n>=1} (-1)^n* a(n)/(A132095(n)*x^(2*n-1)) as x -> oo. - Robert Israel, May 27 2015

Extensions

More terms from R. J. Mathar, Oct 18 2007
Meaningful name from Joerg Arndt, Jan 25 2024

A133002 Numerators of Blandin-Diaz compositional Bernoulli numbers (B^S)_1,n.

Original entry on oeis.org

1, -1, 5, -1, 139, -1, 859, 71, -9769, 233, -6395527, 145069, -304991568097, -95164619917, 119780081383, -3046785293, 4002469707564917, -102407337854027, 1286572077762833639, 219276930957009857, -20109624681057406222913, 1651690537394493957719
Offset: 0

Views

Author

Jonathan Vos Post, Aug 09 2007

Keywords

Comments

Denominators are A133003. "Bernoulli numbers for S are shown in the table."
The signs of a(0) and a(3) are wrong in table of p. 11 of Bandin article. - Daniel Suteu, Feb 24 2018

Examples

			1, -1/4, 5/72, -1/48, 139/21600, -1/540, 859/2540160, 71/483840, -9769/36288000 (corrected by _Daniel Suteu_, Feb 24 2018).

Links

Crossrefs

Cf. A132092, A132093, A132094, A132095, A132096, A132097, A132098, A132099.
Cf. A133000, A133001, A133003, A133004, A133005.

Programs

  • Mathematica
    f[0] = 1; f[n_] := f[n] = -Sum[f[k]/((n-k+1)!)^2, {k, 0, n-1}]; Table[f[n]*n! // Numerator, {n, 0, 21}] (* Jean-François Alcover, Feb 25 2018, after Daniel Suteu *)

Formula

a(n) = numerator(f(n) * n!), where f(0) = 1, f(n) = -Sum_{k=0..n-1} f(k) / ((n-k+1)!)^2. - Daniel Suteu, Feb 23 2018
E.g.f. for fractions: x / (BesselI(0,2*sqrt(x)) - 1). - Ilya Gutkovskiy, Sep 01 2021

Extensions

Corrected the sign of a(0) and a(3) by Daniel Suteu, Feb 24 2018
Terms beyond a(8) from Daniel Suteu, Feb 24 2018

A133003 Denominators of Blandin-Diaz compositional Bernoulli numbers (B^S)_1,n.

Original entry on oeis.org

1, 4, 72, 48, 21600, 540, 2540160, 483840, 36288000, 896000, 31614105600, 1149603840, 7139902049280000, 2196892938240000, 941525544960000, 15216574464000, 16326052949606400000, 443241256550400000, 11991344662654156800000, 1100420292929126400000
Offset: 0

Views

Author

Jonathan Vos Post, Aug 09 2007

Keywords

Comments

Numerators are A133002.

Examples

			1, -1/4, 5/72, -1/48, 139/21600, -1/540, 859/2540160, 71/483840, -9769/36288000 (corrected by _Daniel Suteu_, Feb 24 2018).

Links

Crossrefs

Cf. A132092, A132093, A132094, A132095, A132096, A132097, A132098, A132099.
Cf. A133000, A133001, A133002, A133004, A133005.

Programs

  • Mathematica
    f[0] = 1; f[n_] := f[n] = -Sum[f[k]/((n - k + 1)!)^2, {k, 0, n - 1}]; a[n_] := Denominator[f[n]*n!]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 25 2018, after Daniel Suteu *)

Formula

a(n) = denominator(f(n) * n!), where f(0) = 1, f(n) = -Sum_{k=0..n-1} f(k) / ((n-k+1)!)^2. - Daniel Suteu, Feb 23 2018
E.g.f. for fractions: x / (BesselI(0,2*sqrt(x)) - 1). - Ilya Gutkovskiy, Sep 01 2021

Extensions

Terms beyond a(8) from Daniel Suteu, Feb 24 2018
Showing 1-3 of 3 results.

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