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 1-6 of 6 results.

A132092 Numerators of Blandin-Diaz compositional Bernoulli numbers (B^sin)_3,n.

Original entry on oeis.org

-1, -1, -11, -17, -563, -381, 55277, 242747, 406146379, 104180627, -398489682593, -169622229019, -6523856615663, -251077358513783, 35076901882951197, 2869253069531102351, 20717378005021857058651, 1335883610404565359777223, 27846976637614329871324177
Offset: 0

Views

Author

Jonathan Vos Post, Aug 09 2007

Keywords

Comments

Denominators are A132093. Numerators and denominators given only for even n (odd n have numerators = 0).

References

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

Links

Crossrefs

Cf. A132093 (denominators), A132094-A132099.

Programs

  • Maple
    A132092 := proc(n) local g; g := taylor(sin(x)-x,x=0,n+7) ; g := taylor(g/x^3,x=0,n+4) ; g := taylor( 1/6/g,x=0,n+4) ; n!*coeftayl(g,x=0,n) ; numer(%) ; end: for n from 0 to 40 by 2 do printf("%d,",A132092(n)) ; od: # R. J. Mathar, May 25 2008
  • Mathematica
    m = 20;
((x^3)/3!)/(Sin[x]-x) + O[x]^(2m) // CoefficientList[#, x]& // #*Range[0, 2m-2]!& // #[[;; ;; 2]]& // Numerator (* Jean-François Alcover, Mar 23 2020 *)
  • PARI
    my(N=40, x='x+O('x^N), v=apply(numerator, Vec(serlaplace(x^3/(6*(sin(x)-x)))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Jan 24 2024

Formula

((x^3)/3!)/(sin(x)-x) = Sum_{n>=0} (B^sin)_3,n ((x^n)/n!).

Extensions

More terms from R. J. Mathar, May 25 2008
Offset corrected by Andrew Howroyd, Sep 22 2024

A132095 Denominators of expansion of e.g.f. x^2/(2*(cos(x)-1)), even powers only.

Original entry on oeis.org

1, 6, 10, 42, 30, 22, 2730, 6, 34, 798, 330, 46, 2730, 6, 290, 14322, 510, 2, 54834, 6, 4510, 1806, 690, 94, 46410, 66, 530, 798, 174, 118, 56786730, 6, 170, 64722, 30, 1562, 140100870, 6, 2, 474, 230010, 166, 3404310, 6, 20470, 272118, 1410, 2, 900354, 6
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

  • Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers , Afr. Diaspora J. Math., Volume 7, Number 2 (2008).
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199. See Table 3.3.

Links

Crossrefs

Numerators are A132094.
Cf. A132092-A132099.

Programs

  • Maple
    A132095 := proc(n) add( 2*(-1)^i*x^(2*i)/(2*i+2)!,i=0..n/2+1) ; denom(coeftayl(-1/%,x=0,n)*n!) ; end: for n from 0 to 46 by 2 do printf("%d, ",A132095(n)) ; od: # R. J. Mathar, Oct 18 2007
  • Mathematica
    A132095[n_] := (s = Sum[ 2*(-1)^i*x^(2*i)/(2*i + 2)!, {i, 0, n/2 + 1}] ; Denominator[SeriesCoefficient[-1/s, {x, 0, n}]*n!]) ;
Table[ A132095[n], {n, 0, 100, 2}] (* Jean-François Alcover, Nov 24 2017, after R. J. Mathar *)
  • PARI
    my(x='x+O('x^100), v=apply(denominator, 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* A132094(n)/(a(n)*x^(2*n-1)) as x -> infinity. - Robert Israel, May 27 2015

Extensions

More terms from R. J. Mathar, Oct 18 2007 and Oct 20 2009
Meaningful name from Joerg Arndt, Jan 25 2024

A132093 Denominators of Blandin-Diaz compositional Bernoulli numbers (B^sin)_3,n.

Original entry on oeis.org

1, 10, 350, 1050, 57750, 250250, 2388750, 2231250, 1088106250, 137156250, 105761906250, 2289218750, 8842968750, 51289218750, 45049030468750, 3563716406250, 1099667378906250, 4714260332031250, 14142780996093750
Offset: 0

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, -1/10, 0, -11/350, 0, -17,1050, 0, -563/57750, 0, -381/250250, 0.

References

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

Links

Crossrefs

Numerators are A132092.
Cf. A132094-A132099.

Programs

  • Mathematica
    m = 20;
((x^3)/3!)/(Sin[x]-x) + O[x]^(2m) // CoefficientList[#, x]& // #*Range[0, 2m-2]!& // #[[;; ;; 2]]& // Denominator (* Jean-François Alcover, Mar 23 2020 *)
  • PARI
    my(N=40, x='x+O('x^N), v=apply(denominator, Vec(serlaplace(x^3/(6*(sin(x)-x)))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Jan 24 2024

Formula

((x^3)/3!)/(sin(x)-x) = Sum_{n>=0} (B^sin)_3,n ((x^n)/n!).

Extensions

More terms from R. J. Mathar, May 25 2008
Offset corrected as suggested by Andrew Howroyd. - N. J. A. Sloane, Sep 22 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

A160035 Clausen-normalized numerators of the Bernoulli numbers of order 2.

Original entry on oeis.org

1, 0, -1, 0, 3, 0, -5, 0, 7, 0, -45, 0, 7601, 0, -91, 0, 54255, 0, -745739, 0, 3317609, 0, -17944773, 0, 5436374093, 0, -213827575, 0, 641235447783, 0, -249859397004145, 0, 238988952277727, 0, -85063699326111, 0, 921034504356871708055, 0, -108409774812137683
Offset: 0

Views

Author

Peter Luschny, Apr 30 2009

Keywords

Comments

Let B_n{^(k)}(x) denote the Bernoulli polynomials of order k, defined by the generating function
(t/(exp(t)-1))^k*exp(x*t) = Sum_{n>=0} B_n{^(k)}(x) t^n/n!
Bernoulli numbers of order 1 (defined as B_n{^(1)}(1)) can be regarded as a pair of sequences B1_n = N1_n / D1_n with
N1_n = A027641, D1_n = A141056 (Clausen).
Similarly Bernoulli numbers of order 2 (defined as B_n{^(2)}(1)) can be regarded as a pair of sequences B2_n = N2_n / D2_n with
N2_n = this sequence, D2_n = A141056 (Clausen).

Examples

			The Clausen-normalized Bernoulli polynomials of order 2 are:
1
2 x - 2
6 x^2 - 12 x + 5
2 x^3 - 6 x^2 + 5 x - 1
30 x^4 - 120 x^3 + 150 x^2 - 60 x + 3
2 x^5 - 10 x^4 + 50/3 x^3 - 10 x^2 + x + 1/3
42 x^6 - 252 x^5 + 525 x^4 - 420 x^3 + 63 x^2 + 42 x - 5
The value of these polynomials at x = 1 gives the sequence.

References

  • L. Comtet, Advanced Combinatorics, Reidel, Boston, Mass., 1974.
  • C. Jordan, Calculus of Finite Differences, New York, Chelsea, 1965.
  • N. E. Nørlund, Vorlesungen über Differenzenrechnung, Berlin, Springer-Verlag, 1924.

Crossrefs

Cf. A120282, A132094, A100615 and A027643.

Programs

  • Maple
    aList := proc(n) local g,c,i; g := k -> (t/(exp(t)-1))^k*exp(x*t): c := proc(n) local i; mul(i,i=select(isprime, map(i->i+1,numtheory[divisors](n)))) end: convert(series(g(2),t,n+8),polynom): seq(i!*c(i)*subs(x=1,coeff(%,t,i)),i=0..n) end: aList(38);
  • Mathematica
    aList[n_] := Module[{g, c, s},
   g[k_] := (t/(Exp[t]-1))^k*Exp[x*t];
   c[k_] := Times @@ Select[Divisors[k]+1, PrimeQ];
   s = Series[g[2], {t, 0, n + 8}] // Normal;
Join[{1}, Table[i!*c[i]*Coefficient[s, t, i] /. x -> 1, {i, 1, n}]]];
aList[38] (* Jean-François Alcover, May 28 2024, after Peter Luschny *)
Showing 1-6 of 6 results.

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