A277001 -id:A277001 - cp's OEIS frontend

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

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

Offset: 0

Peter Luschny, Sep 25 2016

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).

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

Peter Luschny, Approximations to the factorial function.
W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

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).

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

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 *)

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

A277002 Numerators of an asymptotic series for the Gamma function (odd power series).

Original entry on oeis.org

-1, 7, -31, 127, -511, 1414477, -8191, 118518239, -5749691557, 91546277357, -23273283019, 1982765468311237, -22076500342261, 455371239541065869, -925118910976041358111, 16555640865486520478399, -1302480594081611886641, 904185845619475242495834469891

Offset: 1

Peter Luschny, Sep 26 2016

Let y = x+1/2 then Gamma(x+1) ~ sqrt(2*Pi)*(y/E)^y*exp(Sum_{k>=1} r(k)/y^(2*k-1)) as x -> oo and r(k) = A277002(k)/A277003(k) (see example 7.1 in the Wang reference).

See also theorem 2 and formula (58) in Borwein and Corless. - Peter Luschny, Mar 31 2017

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

J. M. Borwein, R. M. Corless, Gamma and Factorial in the Monthly, arXiv:1703.05349 [math.HO], 2017.
Peter Luschny, Approximations to the factorial function.
W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

Cf. A277003 (denominators), A277000/A277001 (even power series).

b := n -> `if`(n=0, 0, bernoulli(n+1, 1/2)/(n*(n+1))):
a := n -> numer(b(2*n-1)):
seq(a(n), n=1..18);

b[n_] := BernoulliB[n+1, 1/2]/(n(n+1));
a[n_] := Numerator[b[2n-1]];
Array[a, 18] (* Jean-François Alcover, Sep 09 2018 *)

a(n) = numerator(b(2*n-1)) with b(n) = Bernoulli(n+1, 1/2)/(n*(n+1)) for n>=1, b(0)=0.

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

Peter Luschny, Sep 26 2016

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

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

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

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);

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 *)

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

A276996 Numerators of coefficients of polynomials arising from applying the complete Bell polynomials to k!B_k(x)/(k*(k-1)) with B_k(x) the Bernoulli polynomials.

Original entry on oeis.org

1, 0, 0, 1, -1, 1, 0, 1, -3, 1, 1, -1, 6, -10, 5, 0, -1, -15, 95, -40, 16, 239, -1, 13, -85, 240, -237, 79, 0, 403, 21, 385, -1575, 3577, -2947, 421, -46409, -239, 3841, 175, 861, -8036, 45458, -10692, 2673, 0, -82451, -2657, 56177, 1638, 19488, -85260, 139656, -86472, 19216

Offset: 0

Peter Luschny, Oct 01 2016

The polynomials appear in certain asymptotic series for the Gamma function, cf. for example A181855/A181856 and A277000/A277001.

			Polynomials start:
p_0(x) = 1;
p_1(x) = 0;
p_2(x) = 1/6 + -x + x^2;
p_3(x) = (1/2)*x + -(3/2)*x^2 + x^3;
p_4(x) = 1/60 + -x + 6*x^2 + -10*x^3 + 5*x^4;
p_5(x) = -(1/6)*x + -(15/2)*x^2 + (95/3)*x^3 + -40*x^4 + 16*x^5;
p_6(x) = 239/504 + -(1/4)*x + (13/4)*x^2 + -85*x^3 + 240*x^4 + -237*x^5 + 79*x^6;
Triangle starts:
1;
0,   0;
1,  -1,   1;
0,   1,  -3,   1;
1,  -1,   6, -10,  5;
0,  -1, -15,  95, -40,   16;
239,-1,  13, -85, 240, -237, 79;

W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

Cf. A276997 (denominators); T(2n,0) = A181855(n), T(n,n) = A203852(n).

Cf. A276998.

A276996_row := proc(n) local p;
p := (n,x) -> CompleteBellB(n,0,seq((k-2)!*bernoulli(k,x),k=2..n)):
seq(numer(coeff(p(n,x),x,k)), k=0..n) end:
seq(A276996_row(n), n=0..9);
# Recurrence for the polynomials:
A276996_poly := proc(n,x) option remember; local z;
if n = 0 then return 1 fi; z := proc(k) option remember;
if k=1 then 0 else (k-2)!*bernoulli(k,x) fi end;
expand(add(binomial(n-1,j)*z(n-j)*A276996_poly(j,x),j=0..n-1)) end:
for n from 0 to 5 do sort(A276996_poly(n,x)) od;

CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
p[n_, x_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, x], {k, 2, n}]]];
row[0] = {1}; row[1] = {0, 0}; row[n_] := CoefficientList[p[n, x], x] // Numerator;
Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Sep 09 2018 *)

T(n,k) = Numerator([x^k] p_n(x)) where p_n(x) = Y_{n}(z_1, z_2, z_3,..., z_n) are the complete Bell polynomials evaluated at z_1 = 0 and z_k = (k-2)!*B_k(x) for k>1 and B_k(x) the Bernoulli polynomials.

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

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A277002 Numerators of an asymptotic series for the Gamma function (odd power series).

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

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A276996 Numerators of coefficients of polynomials arising from applying the complete Bell polynomials to k!B_k(x)/(k*(k-1)) with B_k(x) the Bernoulli polynomials.

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