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.

A277000 Numerators of an asymptotic series for the Gamma function (even power series).

Original entry on oeis.org

1, -1, 19, -2561, 874831, -319094777, 47095708213409, -751163826506551, 281559662236405100437, -49061598325832137241324057, 5012066724315488368700829665081, -26602063280041700132088988446735433, 40762630349420684160007591156102493590477
Offset: 0

Views

Author

Peter Luschny, Sep 25 2016

Keywords

Comments

Let y = x+1/2 then Gamma(x+1) ~ sqrt(2*Pi)*((y/E)*Sum_{k>=0} r(k)/y^(2*k))^y as x -> oo and r(k) = A277000(k)/A277001(k) (see example 6.1 in the Wang reference).

Examples

			The underlying rational sequence starts:
1, 0, -1/24, 0, 19/5760, 0, -2561/2903040, 0, 874831/1393459200, 0, ...

Links

Crossrefs

Cf. A001163/A001164 (Stirling), A182935/A144618 (De Moivre), A005146/A005147 (Stieltjes), A090674/A090675 (Lanczos), A181855/A181856 (Nemes), A182912/A182913 (NemesG), A182916/A182917 (Wehmeier), A182919/A182920 (Gosper), A182914/A182915, A277002/A277003 (odd power series).
Cf. A276667/A276668 (the arguments of the Bell polynomials).

Programs

  • Maple
    b := n -> CompleteBellB(n, 0, seq((k-2)!*bernoulli(k,1/2), k=2..n))/n!:
A277000 := n -> numer(b(2*n)): seq(A277000(n), n=0..12);
# Alternatively the rational sequence by recurrence:
R := proc(n) option remember; local k; `if`(n=0, 1,
add(bernoulli(2*m+2,1/2)* R(n-m-1)/(2*m+1), m=0..n-1)/(2*n)) end:
seq(numer(R(n)), n=0..12); # Peter Luschny, Sep 30 2016
  • Mathematica
    CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
b[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1/2], {k, 2, n}]]]/n!;
a[n_] := Numerator[b[2n]];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Sep 09 2018 *)

Formula

a(n) = numerator(b(2*n)) with b(n) = Y_{n}(0, z_2, z_3,..., z_n)/n! with z_k = k!*Bernoulli(k,1/2)/(k*(k-1)) and Y_{n} the complete Bell polynomials.
The rational numbers have the recurrence r(n) = (1/(2*n))*Sum_{m=0..n-1} Bernoulli(2*m+2,1/2)*r(n-m-1)/(2*m+1) for n>=1, r(0)=1. - Peter Luschny, Sep 30 2016

A277001 Denominators of an asymptotic series for the Gamma function (even power series).

Original entry on oeis.org

1, 24, 5760, 2903040, 1393459200, 367873228800, 24103053950976000, 115694658964684800, 9440684171518279680000, 271211974879377138647040000, 3579998068407778230140928000000, 1976158933761093583037792256000000, 258955866680053703121272297226240000000
Offset: 0

Views

Author

Peter Luschny, Sep 25 2016

Keywords

Comments

For formulas and references see A277000 which is the main entry for this rational sequence.

Examples

			The underlying rational sequence starts:
1, 0, -1/24, 0, 19/5760, 0, -2561/2903040, 0, 874831/1393459200, 0, ...

Crossrefs

Cf. A277000 (numerators), A277002/A277003 (odd power series).

Programs

  • Maple
    b := n -> CompleteBellB(n,0,seq((k-2)!*bernoulli(k,1/2),k=2..n))/n!:
A277001 := n -> denom(b(2*n)): seq(A277001(n), n=0..12);
  • Mathematica
    CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
b[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1/2], {k, 2, n}]]]/n!;
a[n_] := Denominator[b[2n]];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Sep 09 2018 *)

A277003 Denominators of an asymptotic series for the Gamma function (odd power series).

Original entry on oeis.org

24, 2880, 40320, 215040, 608256, 738017280, 1277952, 4010803200, 32006209536, 65745715200, 1736441856, 12641296711680, 10066329600, 12611097722880, 1337897345089536, 1086454927196160, 3401614098432, 83088011510887219200, 61022895341568
Offset: 1

Views

Author

Peter Luschny, Sep 26 2016

Keywords

Comments

For formulas and references see A277002 which is the main entry for this rational sequence.

Examples

			The underlying rational sequence b(n) starts:
0, -1/24, 0, 7/2880, 0, -31/40320, 0, 127/215040, 0, -511/608256, ...

Crossrefs

Cf. A277002 (numerators), A277000/A277001 (even power series).

Programs

  • Maple
    b := n -> `if`(n=0, 0, bernoulli(n+1, 1/2)/(n*(n+1))):
a := n -> denom(b(2*n-1)):
seq(a(n), n=1..19);
  • Mathematica
    b[n_] := BernoulliB[n+1, 1/2]/(n(n+1));
a[n_] := Denominator[b[2n-1]];
Array[a, 19] (* Jean-François Alcover, Sep 09 2018 *)

Formula

a(n) = denominator(b(2*n-1)) with b(n) = Bernoulli(n+1, 1/2)/(n*(n+1)) for n>=1, b(0)=0.
Showing 1-3 of 3 results.

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