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 1-4 of 4 results.

A132099 Denominators of Blandin-Diaz compositional Bernoulli numbers (B^Z^2)_1,n.

Original entry on oeis.org

1, 8, 432, 144, 324000, 64800, 16669800
Offset: 0

Views

Author

Jonathan Vos Post, Aug 09 2007

Keywords

Examples

			1, -1/8, 11/432, 1/144, -217/324000, -157/64800, -21503/16669800.

Links

Crossrefs

Numerators are A132098.
Cf. A132092-A132097.

A132096 Numerators of Blandin-Diaz compositional Bernoulli numbers (B^Z)_1,n.

Original entry on oeis.org

1, -1, 1, 1, 61, -1, -12491, -479, 530629, 54979, 1039405, -4981183, -9055875786121, 908993573959, 288260975797477, 7874837285353, -2255621632465386299, -189404901989770501, -20038592583515962234111, 954329155426992424481, 1731149375200514221429374109
Offset: 0

Views

Author

Jonathan Vos Post, Aug 09 2007

Keywords

Examples

			1, -1/4, 1/72, 1/96, 61/21600, -1/640, -12491/5080320, -479/680608.

Links

Crossrefs

Denominators are A132097.
Cf. A132092-A132099.

Programs

  • Mathematica
    nn = 21; A = Inverse[Table[Table[If[n >= k, Binomial[n - 1, k - 1]/(n - k + 1)^2, 0], {k, 1, nn}], {n, 1, nn}]]; Numerator[A[[All, 1]]] (* Mats Granvik, Feb 05 2018 *)

Formula

This sequence appears to be the numerators of the first column in the matrix inverse of the lower triangular matrix: If n >= k then binomial(n-1,k-1)/(n-k+1)^2, otherwise 0. - Mats Granvik, Feb 05 2018
a(n) = numerator(f(n)), where f(0) = 1, f(n) = -Sum_{k=0..n-1} f(k) * binomial(n,k) / (n-k+1)^2. - Daniel Suteu, Feb 23 2018

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 1-4 of 4 results.

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