A182913 -id:A182913 - 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

A182912 Numerators of an asymptotic series for the Gamma function (G. Nemes).

Original entry on oeis.org

1, 0, 1, -1, -257, -53, 5741173, 37529, -710165119, -3376971533, 360182239526821, 104939254406053, -508096766056991140541, -70637580369737593, 289375690552473442964467, 796424971106808496421869

Offset: 0

Peter Luschny, Feb 09 2011

G_n = A182912(n)/A182913(n). These rational numbers provide the coefficients for an asymptotic expansion of the Gamma function.

			G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -1/12960.

G. Nemes, More Accurate Approximations for the Gamma Function, Thai Journal of Mathematics Volume 9(1) (2011), 21-28.

Peter Luschny, Approximation Formulas for the Factorial Function.

Cf. A001163, A001164, A182913.

G := proc(n) option remember; local j,J;
J := proc(k) option remember; local j; `if`(k=0,1,
(J(k-1)/k-add((J(k-j)*J(j))/(j+1),j=1..k-1))/(1+1/(k+1))) end:
add(J(2*j)*2^j*6^(j-n)*GAMMA(1/2+j)/(GAMMA(n-j+1)*GAMMA(1/2+j-n)),j=0..n)-add(G(j)*(-4)^(j-n)*(GAMMA(n)/(GAMMA(n-j+1)*GAMMA(j))),j=1..n-1) end:
A182912 := n -> numer(G(n)); seq(A182912(i),i=0..15);

G[n_] := G[n] = Module[{j, J}, J[k_] := J[k] = Module[{j}, If[k == 0, 1, (J[k-1]/k - Sum[J[k-j]*J[j]/(j+1), {j, 1, k-1}])/(1+1/(k+1))]]; Sum[J[2*j]*2^j*6^(j-n)*Gamma[1/2+j]/(Gamma[n-j+1]*Gamma[1/2+j-n]), {j, 0, n}] - Sum[G[j]*(-4)^(j-n)*Gamma[n]/(Gamma[n-j+1]*Gamma[j]), {j, 1, n-1}]]; A182912[n_] := Numerator[G[n]]; Table[A182912[i], {i, 0, 15}] (* Jean-François Alcover, Jan 06 2014, translated from Maple *)

Gamma(x+1) ~ x^x e^(-x) sqrt(2Pi(x+1/6)) Sum_{n>=0} G_n / (x+1/4)^n.

cp's OEIS Frontend

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

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