A001164 -id:A001164 - cp's OEIS frontend

A257634 a(n) = (A001163(n)/A001164(n))3(2*n)!^2/n!!.

3, 1, 3, -1390, -139895, 2064875400, 999912530925, -128585633463727440, -176876516433064573125, 109242473594498195269718400, 333170810414553853376721961875, -698025623281503752808511373154720000, -4073023833462008382211035330291042675375

Offset: 0

Vladimir Reshetnikov, Nov 04 2015

sign

Coefficients in Stirling's asymptotic expansion of the Gamma function, normalized to integers using factor 3*(2*n)!^2/n!!.

G. C. Greubel, Table of n, a(n) for n = 0..30
Eric Weisstein's World of Mathematics, Stirling's Series.

Cf. A001163, A001164.

h := proc(k) option remember; local j; `if`(k=0,1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1),j=1..k-1))/(1+1/(k+1))) end:
g := n -> doublefactorial(2*n-1)*(2*n)!^2/doublefactorial(n):
seq(3*h(2*n)*g(n), n=0..12); # Peter Luschny, Nov 05 2015

Table[3 (2n)!^2/n!! (6n+1)!!/4^n Sum[(-1)^m 2^k StirlingS2[2n+k+m, m]/((2n+2k+1) (2n+k+m)! (2n-k)! (k-m)!), {k, 0, 2n}, {m, 0, k}], {n, 0, 12}]

a(n) = 3*(2*n)!*(6*n+1)!!/(n!!*4^n) * Sum_{i=0..2*n} Sum_{j=0..i} Sum_{k=0..j} (-1)^k*2^i*k^(2*n+i+j)*C(2*n,i)*C(i,j)*C(j,k) / ((2*n+2*i+1)*(2*n+i+j)!), assuming 0^0 = 1 (when n = 0), n!! = A006882(n), C(n,k) = A007318(n,k) are binomial coefficients.

A001163 Stirling's formula: numerators of asymptotic series for Gamma function.

Original entry on oeis.org

1, 1, 1, -139, -571, 163879, 5246819, -534703531, -4483131259, 432261921612371, 6232523202521089, -25834629665134204969, -1579029138854919086429, 746590869962651602203151, 1511513601028097903631961, -8849272268392873147705987190261, -142801712490607530608130701097701

Offset: 0

N. J. A. Sloane

			Gamma(z) ~ sqrt(2*Pi) * z^(z-1/2) * e^(-z) * (1 + 1/(12*z) + 1/(288*z^2) - 139/(51840*z^3) - 571/(2488320*z^4) + ... ), z -> infinity in |arg z| < Pi.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.37.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 267, #23.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:6:1 at page 21.

Seiichi Manyama, Table of n, a(n) for n = 0..227 (terms 0..100 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.37.
S. Brassesco and M. A. Méndez, The asymptotic expansion for the factorial and Lagrange inversion formula, arXiv:1002.3894 [math.CA], 2010.
V. De Angelis, Stirling's series revisited, Amer. Math. Monthly, 116 (2009), 839-843.
N. Elezovic, Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers, J. Int. Seq. 17 (2014) # 14.2.1
Peter Luschny, Approximations to the factorial function.
G. Marsaglia and J. C. W. Marsaglia, A new derivation of Stirling's approximation to n!, Amer. Math. Monthly, 97 (1990), 827-829. MR1080390 (92b:41049)
T. Mueller, Finite group actions and asymptotic expansion of e^P(z), Combinatorica, 17 (4) (1997), 523-554.
Richard M. Slevinsky, On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev-Jacobi transform, arXiv preprint arXiv:1602.02618 [math.NA], 2016.
N. M. Temme, The asymptotic expansion of the incomplete gamma function, SIAM J. Math. Anal., 10 (1979), 757-766. [From _N. J. A. Sloane_, Feb 20 2012]
W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).
Eric Weisstein's World of Mathematics, Stirling's Series.
J. W. Wrench, Jr., Concerning two series for the gamma function, Math. Comp., 22 (1968), 617-626.

Cf. A001164 (denominators).
Cf. A097303 (see W. Lang link there for a similar numerator sequence which deviates for the first time at entry number 33. Expansion of GAMMA(z) in terms of 1/(k!*z^k) instead of 1/z^k).
Product_{z=1..n} z^(z^m): A143475/A143476 (m=1), A317747/A317796 (m=2), A318713/A318714 (m=3).

h := proc(k) option remember; local j; `if`(k=0, 1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
StirlingAsympt := proc(n) option remember; h(2*n)*2^n*pochhammer(1/2, n) end:
A001163 := n -> numer(StirlingAsympt(n));
seq(A001163(n), n=0..30); # Peter Luschny, Feb 08 2011

Numerator[ Reverse[ Drop[ CoefficientList[ Simplify[ PowerExpand[ Normal[ Series[n!, {n, Infinity, 17}]]Exp[n]/(Sqrt[2Pi n]*n^(n - 17))]], n], 1]]]
(* Second program: *)
h[k_] := h[k] = If[k==0, 1, (h[k-1]/k-Sum[h[k-j]*h[j]/(j+1), {j, 1, k-1}]) / (1+1/(k+1))];
StirlingAsympt[n_] := h[2n]*2^n*Pochhammer[1/2, n];
a[n_] := StirlingAsympt[n] // Numerator;
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 12 2015, after Peter Luschny *)

a(n)=local(A,m); if(n<1,n==0,A=vector(m=2*n+1,k,1); for(k=2,m,A[k]=(A[k-1]-sum(i=2,k-1,i*A[i]*A[k+1-i]))/(k+1)); numerator(A[m]*m!/2^n/n!)) /* Michael Somos, Jun 09 2004 */

def A001163(n):
    @cached_function
    def h(k):
        if k<=0: return 1
        S = sum((h(k-j)*h(j))/(j+1) for j in (1..k-1))
        return (h(k-1)/k-S)/(1+1/(k+1))
    return numerator(h(2*n)*2^n*rising_factorial(1/2,n))
[A001163(n) for n in range(17)] # Peter Luschny, Nov 05 2015

The coefficients c_k have g.f. 1 + Sum_{k >= 1} c_k/z^k = exp( Sum_{k >= 1} B_{2k}/(2k(2k-1)z^(2k-1)) ).
Numerators/denominators: A001163(n)/A001164(n) = (6*n+1)!!/(4^n*(2*n)!) * Sum_{i=0..2*n} Sum_{j=0..i} Sum_{k=0..j} (-1)^k*2^i*k^(2*n+i+j)*C(2*n,i) *C(i,j)*C(j,k)/((2*n+2*i+1)*(2*n+i+j)!), assuming 0^0 = 1 (when n = 0), n!! = A006882(n), C(n,k) = A007318(n,k) are binomial coefficients. - Vladimir Reshetnikov, Nov 05 2015
From Seiichi Manyama, Sep 01 2018: (Start)
Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (1/n) * Sum_{k=0..n-1} B_{n-k+1}*c_k/(n-k+1) for n > 0.
a(n) is the numerator of c_n. (End)

More terms from Vladeta Jovovic, Nov 14 2001

Signs added by Robert G. Wilson v, Jul 12 2003

A144618 Denominators of an asymptotic series for the factorial function (Stirling's formula with half-shift).

Original entry on oeis.org

1, 24, 1152, 414720, 39813120, 6688604160, 4815794995200, 115579079884800, 22191183337881600, 263631258054033408000, 88580102706155225088000, 27636992044320430227456000, 39797268543821419527536640000

Offset: 0

N. J. A. Sloane, Jan 15 2009, based on email from Chris Kormanyos (ckormanyos(AT)yahoo.com)

From Peter Luschny, Feb 24 2011 (Start):
G_n = A182935(n)/A144618(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function.
The relationship between these coefficients and the Bernoulli numbers are due to De Moivre, 1730 (see Laurie). (End)
Also denominators of polynomials mentioned in A144617.
Also denominators of polynomials mentioned in A144622.

			G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.

Chris Kormanyos, Table denominators of u_k for k=0..121
Dirk Laurie, Old and new ways of computing the gamma function, page 14, 2005.
Peter Luschny, Approximation Formulas for the Factorial Function.
W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

Cf. A001163, A001164, A182935, A144617, A144622.

G := proc(n) option remember; local j,R;
R := seq(2*j,j=1..iquo(n+1,2));
`if`(n=0,1,add(bernoulli(j,1/2)*G(n-j+1)/(n*j),j=R)) end:
A144618 := n -> denom(G(n)); seq(A144618(i),i=0..12);
# Peter Luschny, Feb 24 2011

a[0] = 1; a[n_] := a[n] = Sum[ BernoulliB[j, 1/2]*a[n-j+1]/(n*j), {j, 2, n+1, 2}]; Table[a[n] // Denominator, {n, 0, 12}] (* Jean-François Alcover, Jul 26 2013, after Maple *)

z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2) Sum_{n>=0} G_n / (z+1/2)^n.

- Peter Luschny, Feb 24 2011

Added more terms up to polynomial number u_12, v_12 for the denominators of u_k, v_k. Christopher Kormanyos (ckormanyos(AT)yahoo.com), Jan 31 2009
Typo in definition corrected Aug 05 2010 by N. J. A. Sloane
A-number in definition corrected - R. J. Mathar, Aug 05 2010
Edited and new definition by Peter Luschny, Feb 24 2011

A317747 Numerator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

Original entry on oeis.org

1, -1, 1, 259193, -1036793, -201551328007, 9137074752049, 9142431862033871923, -11105299580705049589, -11003865617473929216508154207, 114467620015003245418244743007, 32505236416490926096399421788847363, -254505521478572052318535393350091231, -1828472168539763642032546635313363411876021

Offset: 0

Seiichi Manyama, Sep 01 2018

1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_2 is the second Bendersky constant.

a(n) is the numerator of b(n).

			1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(1 - 1/(360*n) + 1/(259200*n^2) + 259193/(1959552000*n^3) - 1036793/(2821754880000*n^4) - 201551328007/(5079158784000000*n^5) + ... ).

Seiichi Manyama, Table of n, a(n) for n = 0..174
Weiping Wang, Some asymptotic expansions on hyperfactorial functions and generalized Glaisher-Kinkelin constants, ResearchGate, 2017.

Product_{z=1..n} z^(z^m): A001163/A001164 (m=0), A143475/A143476 (m=1), A317747/A317796 (m=2).

Cf. A051675, A243262 (A_2).

Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (2/n) * Sum_{k=0..n-1} B_{n-k+3}*c_k/((n-j+1)*(n-k+2)*(n-k+3)) for n > 0.
a(n) is the numerator of c_n.

A317796 Denominator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

Original entry on oeis.org

1, 360, 259200, 1959552000, 2821754880000, 5079158784000000, 76796880814080000000, 304115648023756800000000, 125121866615488512000000000, 258236518070374430146560000000000, 929651465053347948527616000000000000, 334674527419205261469941760000000000000, 920050700832433373350094438400000000000000

Offset: 0

Seiichi Manyama, Sep 01 2018

1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_2 is the second Bendersky constant.

a(n) is the denominator of b(n).

			1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(1 - 1/(360*n) + 1/(259200*n^2) + 259193/(1959552000*n^3) - 1036793/(2821754880000*n^4) - 201551328007/(5079158784000000*n^5) + ... ).

Seiichi Manyama, Table of n, a(n) for n = 0..206
Weiping Wang, Some asymptotic expansions on hyperfactorial functions and generalized Glaisher-Kinkelin constants, ResearchGate, 2017.

Product_{z=1..n} z^(z^m): A001163/A001164 (m=0), A143475/A143476 (m=1), A317747/A317796 (m=2).

Cf. A051675, A243262 (A_2).

Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (2/n) * Sum_{k=0..n-1} B_{n-k+3}*c_k/((n-j+1)*(n-k+2)*(n-k+3)) for n > 0.
a(n) is the denominator of c_n.

A006488 Numbers n such that n! has a square number of digits.

Original entry on oeis.org

0, 1, 2, 3, 7, 12, 18, 32, 59, 81, 105, 132, 228, 265, 284, 304, 367, 389, 435, 483, 508, 697, 726, 944, 1011, 1045, 1080, 1115, 1187, 1454, 1494, 1617, 1659, 1788, 1921, 2012, 2105, 2200, 2248, 2395, 2445, 2861, 2915, 3192, 3480, 3539, 3902, 3964, 4476

Offset: 1

N. J. A. Sloane and Robert G. Wilson v

Numbers whose square is represented by the number of digits of n!: 1, 2, 3, 4, 6, 9, 11, 13, 15, 21, 23, 24, 25, 28, 29, ..., . - Robert G. Wilson v, May 14 2014
From Bernard Schott, Jan 04 2020: (Start)
In M. Gardner's book, see reference, there is a tree printout of 105! with 169 digits, where the bottom row consists of the 25 trailing zeros of 105!. M. Gardner does not explain if this is the only factorial that can be displayed in a similar tree form.
Proof: If m! has q^2 digits, hence the number of trailing zeros in m! must be equal to 2*q-1 to satisfy this triangular look; m = 105 satisfies these two conditions with q = 13 because 105! has 13^2 = 169 digits and 2*13-1 = 25 trailing zeros.
When m < 105 and m! has q^2 digits (m <= 81), then q <= 11 and the number of trailing zeros is <= 2*q - 3.
When m > 105 and m! has q^2 digits (m >= 132), then q >= 15 and the number of trailing zeros is >= 2*q + 2.
Hence, only 105! presents such a tree printout.
              1
             081
            39675
           8240290
          900504101
         30580032964
        9720646107774
       902579144176636
      57322653190990515
     3326984536526808240
    339776398934872029657
   99387290781343681609728
  0000000000000000000000000
(End)

M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 55.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..1311
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979
Eric Weisstein's World of Mathematics, Stirling's Approximation and Stirling's Series
Index entries for sequences related to factorial numbers

Cf. A000142, A027868 (trailing zeros), A034886 (number of digits), A056851.

[k:k in [0..5000]| IsSquare(#Intseq(Factorial(k)))]; // Marius A. Burtea, Jan 04 2020

LogBase10Stirling[n_] := Floor[Log[10, 2 Pi n]/2 + n*Log[10, n/E] + Log[10, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5)]]; Select[ Range[ 4500], IntegerQ[ Sqrt[ (LogBase10Stirling[ # ] + 1)]] & ] (* The Mathematica coding comes from J. Stirling's expansion for the Gamma function; see the links. For more terms inside the last Log_10 function, use A001163 & A001164. Robert G. Wilson v, Apr 27 2014 *)
Select[Range[0,4500],IntegerQ[Sqrt[IntegerLength[#!]]]&] (* Harvey P. Dale, Sep 27 2018 *)

isok(n) = issquare(#Str(n!)); \\ Michel Marcus, Sep 05 2015

A264148 Numerators of rational coefficients related to Stirling's asymptotic series for the Gamma function.

Original entry on oeis.org

1, 2, 1, -4, 1, 8, -139, 16, -571, -8992, 163879, -334144, 5246819, 698752, -534703531, 23349012224, -4483131259, -1357305243136, 432261921612371, -6319924923392, 6232523202521089, 8773495082018816, -25834629665134204969, 49004477022654464, -1579029138854919086429

Offset: 0

Peter Luschny, Nov 05 2015

The rational numbers SGGS = A264148/A264149 (SGGS stands for 'Stirling Generalized Gamma Series') are a supersequence of the coefficients in Stirling's asymptotic series for the Gamma function A001163/A001164 and of an asymptotic expansion of Ramanujan A090804/A065973, further they appear in scaled form in an expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function A005447/A005446.

Ramanujan's asymptotic expansion theta(n) = 1/3+4/(135n)-8/(2835n^2)- ... is considered in the literature also in the form 1-theta(n) (see for example formula (5) in the Choi link). It is this form to which we refer here.

G. C. Greubel, Table of n, a(n) for n = 0..360
K. P. Choi, On the medians of gamma distributions and an equation of Ramanujan, Proceedings of the American Mathematical Society 121:1 (May, 1994), pp. 245-251. [From Vladimir Reshetnikov]
G. Nemes, On the coefficients of the asymptotic expansion of n!, J. Integer Seqs. 13 (2010), 5. [From Vladimir Reshetnikov]

A264148(n) = numerator(SGGS(n)).
A264149(n) = denominator(SGGS(n)).
A001163(n) = numerator(SGGS(2*n)) = numerator(SGGS(2*n)/2^(n+1)).
A001164(n) = denominator(SGGS(2*n)).
A090804(n) = numerator(SGGS(2*n+1)).
A065973(n) = denominator(SGGS(2*n+1)) = denominator(SGGS(2*n+1)/2^(n+1)).
A005447(n+1) = numerator(SGGS(n)/2^(n+1)).
A264150(n) = numerator(SGGS(2*n+1)/2^(n+1)).

h := proc(k) option remember; local j; `if`(k<=0, 1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
SGGS := n -> h(n)*doublefactorial(n-1):
A264148 := n -> numer(SGGS(n)): seq(A264148(n), n=0..24);

h[k_] := h[k] = If[k <= 0, 1, (h[k - 1]/k - Sum[h[k - j]*h[j]/(j + 1), {j, 1, k - 1}]) / (1 + 1/(k + 1))]; a[n_] := h[n]* Factorial2[n - 1] // Numerator; Table[a[n], {n, 0, 24}]

def A264148(n):
    @cached_function
    def h(k):
        if k<=0: return 1
        S = sum((h(k-j)*h(j))/(j+1) for j in (1..k-1))
        return (h(k-1)/k-S)/(1+1/(k+1))
    return numerator(h(n)*(n-1).multifactorial(2))
print([A264148(n) for n in (0..17)])

Let SGGS(n) = h(n)*doublefactorial(n-1) where h(n) = 1 for n<=0 and for n>0 defined by the recurrence (h(k-1)/k - Sum_{j=1..k-1}((h(k-j)*h(j))/(j+1))/ (1+1/(k+1))) then a(n) = numerator(SGGS(n)).

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

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

A063979 Number of decimal digits in (n!)!; A000197.

Original entry on oeis.org

1, 1, 1, 3, 24, 199, 1747, 16474, 168187, 1859934, 22228104, 286078171, 3949867548, 58284826485, 915905054360, 15276520209206, 269617872744249, 5021159048900643, 98417586560408168, 2025488254833817394, 43675043585825292775, 984729344827900257489, 23172929656443132617906

Offset: 0

Robert G. Wilson v, Sep 05 2001

Jon E. Schoenfield, Table of n, a(n) for n = 0..448 (first 100 terms from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Factorial

Cf. A000197, A063944, A034886.

// Using about 100 more digits of precision than needed.
nMax:=30; SetDefaultRealField(RealField(Ceiling(Log(10,Factorial(nMax))+100))); a:=[]; for n in [0..nMax] do a[n+1]:=1+Floor(LogGamma(Factorial(n)+1)/Log(10)); end for; a; // Jon E. Schoenfield, Aug 07 2015

seq(length((n)!!), n=0..19); # Zerinvary Lajos, Mar 10 2007

LogBase10Stirling[n_] := Floor[ Log[10, 2 Pi n]/2 + n*Log[10, n/E] + Log[10, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5) + 5246819/(75246796800n^6)]]; (* A001163/A001164; good to at least a(1000) *) LogBase10Stirling[0] = LogBase10Stirling[1] = 0; Table[1 + LogBase10Stirling[n!], {n, 0, 101}] (* Robert G. Wilson v, Aug 05 2015 *)

\\ Using 100 digits of precision.
 a(n)=localprec(100); my(t=n!);return(floor((t*log(t)-t+1/2*log(2*Pi*t)+1/(12*t))/log(10)+1))\\ Robert Gerbicz, Jul 08 2008

More terms from Vladeta Jovovic, Sep 06 2001
A correspondent reported that terms a(17) - a(19) shown here were wrong. That's not true, they are correct. The correspondent was using Python, where the default precision was not large enough to calculate these terms correctly. Thanks to Brendan McKay, Max Alekseyev and Robert Gerbicz for confirming the entries. - N. J. A. Sloane, Jul 08 2008
a(20) from Brendan McKay, Jul 08 2008
a(21)-a(22) from Hugo Pfoertner, Nov 25 2023

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

A257634 a(n) = (A001163(n)/A001164(n))*3*(2*n)!^2/n!!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A001163 Stirling's formula: numerators of asymptotic series for Gamma function.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

Extensions

A144618 Denominators of an asymptotic series for the factorial function (Stirling's formula with half-shift).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A317747 Numerator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A317796 Denominator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A006488 Numbers n such that n! has a square number of digits.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A264148 Numerators of rational coefficients related to Stirling's asymptotic series for the Gamma function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A257634 a(n) = (A001163(n)/A001164(n))3(2*n)!^2/n!!.