A065973 -id:A065973 - cp's OEIS frontend

A005446 Denominators of expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function.

1, 1, 3, 36, 270, 4320, 17010, 5443200, 204120, 2351462400, 1515591000, 2172751257600, 354648294000, 10168475885568000, 7447614174000, 1830325659402240000, 1595278956070800000, 2987091476144455680000

Offset: 0

N. J. A. Sloane

See A299430/A299431 for more formulas; given g.f. A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n, then A(x)^2 = Sum_{n>=0} A299430(n)/A299431(n)*x^n.

			1, 1/3, 1/36, -1/270, 1/4320, 1/17010, -139/5443200, 1/204120, -571/2351462400, ...
G.f.: A(x) = 1 + x + 1/3*x^2 + 1/36*x^3 - 1/270*x^4 + 1/4320*x^5 + 1/17010*x^6 - 139/5443200*x^7 + 1/204120*x^8 + ... + A005447(n)/A005446(n)x^n + ...

E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 221.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..100
J. M. Borwein and R. M. Corless, Emerging Tools for Experimental Mathematics, Amer. Math. Monthly, 106 (No. 10, 1999), 889-909.
G. Marsaglia and J. C. W. Marsaglia, A new derivation of Stirling's approximation to n!, Amer. Math. Monthly, 97 (1990), 827-829.
J. C. W. Marsaglia, The incomplete gamma function and Ramanujan's rational approximation to exp(x), J. Statist. Comput. Simulation, 24 (1986), 163-168. [_N. J. A. Sloane_, Jun 23 2011]

Cf. A005447, A090804/A065973.

Cf. A299430 / A299431 (A(x)^2), A299432 / A299433.

Maple program from N. J. A. Sloane, Jun 23 2011, based on J. Marsaglia's 1986 paper:
a[1]:=1;
M:=25;
for n from 2 to M do
t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k],k=2..floor(n/2));
if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;
a[n]:=t1;
od:
s1:=[seq(a[n],n=1..M)];

terms = 18; Assuming[x > 0, -ProductLog[-1, -Exp[-1 - x^2/2]] + O[x]^terms] // CoefficientList[#, x]& // Take[#, terms]& // Denominator (* Jean-François Alcover, Jun 20 2013, updated Feb 21 2018 *)

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

a(n)=if(n<1,n==0,denominator(polcoeff(serreverse(sqrt(2*(x-log(1+x+x^2*O(x^n))))),n))) /* Michael Somos, Jun 09 2004 */

@CachedFunction
def b(n): return 1 if (n<2) else (1/(n+1))*( b(n-1) - sum( j*b(n-j+1)*b(j) for j in range(2,n) ))
def A005446(n): return denominator((-1)^n*b(n))
[A005446(n) for n in range(31)] # G. C. Greubel, Nov 21 2022

G.f.: A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n satisfies log(A(x)) = A(x) - 1 - x^2/2.

a(n) = denominator of ((-1)^n * b(n)), where b(n) = (1/(n+1))*( b(n-1) - Sum_{j=2..n-1} j*b(j)*b(n-j+1) ) with b(0) = b(1) = 1 (from Borwein and Corless). - G. C. Greubel, Nov 21 2022

Edited by Michael Somos, Jul 21 2002

A005447 Numerators of the expansion of -W_{-1}(-e^(-1 - x^2/2)) where x > 0 and W_{-1} is the Lambert W function.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, -139, 1, -571, -281, 163879, -5221, 5246819, 5459, -534703531, 91207079, -4483131259, -2650986803, 432261921612371, -6171801683, 6232523202521089, 4283933145517, -25834629665134204969, 11963983648109

Offset: 0

N. J. A. Sloane

Numerators of the expansion of -W_0(-e^(-1 - x^2/2)) where x < 0 an W_0 is the principal branch of the Lambert W function. - Michael Somos, Oct 06 2017

See A299430/A299431 for more formulas; given g.f. A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n, then A(x)^2 = Sum_{n>=0} A299430(n)/A299431(n)*x^n.

			G.f.: A(x) = 1 + x + (1/3)*x^2 + (1/36)*x^3 - (1/270)*x^4 + (1/4320)*x^5 + (1/17010)*x^6 - (139/5443200)*x^7 + (1/204120)*x^8 + ... + (A005447(n)/A005446(n))*x^n + ...

E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 221.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..100
J. M. Borwein and R. M. Corless, Emerging Tools for Experimental Mathematics, Amer. Math. Monthly, 106 (No. 10, 1999), 889-909.
G. Marsaglia and J. C. W. Marsaglia, A new derivation of Stirling's approximation to n!, Amer. Math. Monthly, 97 (1990), 827-829.
NIST Digital Library of Mathematical Functions, Lambert W-Function, section 4.13.7
J. C. W. Marsaglia, The incomplete gamma function and Ramanujan's rational approximation to exp(x), J. Statist. Comput. Simulation, 24 (1986), 163-168. [_N. J. A. Sloane_, Jun 23 2011]

Cf. A005446 (denominators), A090804/A065973.

Cf. A299430 / A299431 (A(x)^2), A299432 / A299433.

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:
A005447 := n -> `if`(n<4,1,`if`(n=4,-1,numer(h(n-1))));
seq(A005447(i),i=0..24); # Peter Luschny, Feb 08 2011
# other program
a[1]:=1;
M:=25;
for n from 2 to M do
t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k],k=2..floor(n/2));
if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;
a[n]:=t1;
od:
s1:=[seq(a[n],n=1..M)]; # N. J. A. Sloane, Jun 23 2011, based on J. Marsaglia's 1986 paper

terms = 25; Assuming[x > 0, -ProductLog[-1, -Exp[-1 - x^2/2]] + O[x]^terms] // CoefficientList[#, x]& // Take[#, terms]& // Numerator (* Jean-François Alcover, Jun 20 2013, updated Feb 21 2018 *)
a[ n_] := If[ n < 0, 0, Block[ {$Assumptions = x < 0}, SeriesCoefficient[ -ProductLog[ -Exp[-1 - x^2/2]], {x, 0, n}] // Numerator]]; (* Michael Somos, Oct 06 2017 *)

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

{a(n) = if( n<1, n==0, numerator( polcoeff( serreverse(sqrt( 2 * (x - log(1 + x + x^2 * O(x^n))))), n)))}; /* Michael Somos, Jun 09 2004 */

@CachedFunction
def h(n): return 1 if (n<1) else ((n+1)/(n+2))*( h(n-1)/n - sum( h(n-j)*h(j)/(j+1) for j in range(1,n) ))
def A005447(n):
    if (n<4): return 1
    elif (n==4): return -1
    else: return numerator(h(n-1))
[A005447(n) for n in range(31)] # G. C. Greubel, Nov 21 2022

G.f.: A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n satisfies log(A(x)) = A(x) - 1 - x^2/2.

Edited by Michael Somos, Jul 21 2002

A090804 Numerators in an asymptotic expansion of Ramanujan.

Original entry on oeis.org

2, -4, 8, 16, -8992, -334144, 698752, 23349012224, -1357305243136, -6319924923392, 8773495082018816, 49004477022654464, -1709650943378038784, -480380831834367035260928, 88481173388026066736939008, 660883915180095254454665216

Offset: 0

N. J. A. Sloane, Feb 11 2004

2^(n+1) divides a(n). - Peter Luschny, Nov 05 2015

			2/3, -4/135, 8/2835, 16/8505, -8992/12629925, -334144/492567075, 698752/1477701225, ...

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge, 1999; Problem 4, p. 616.
B. C. Berndt, Ramanujan's Notebooks II, Springer, 1989; p. 181, Entry 48. See also pp. 184, 193ff.
E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford Univ. Press, 1935; see p. 230, Problem 18.
S. Ramanujan, Collected Papers, edited by G. H. Hardy et al., Cambridge, 1927, pp. 323-324, Question 294.

G. C. Greubel, Table of n, a(n) for n = 0..160
J. D. Buckholtz, Concerning an approximation of Copson, Proc. Amer. Math. Soc., 14 (1963), 564-568.
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.
J. C. W. Marsaglia, The incomplete gamma function and Ramanujan's rational approximation to exp(x), J. Statist. Comput. Simulation, 24 (1986), 163-168. [From _N. J. A. Sloane_, Jun 23 2011]
Cormac O'Sullivan, Ramanujan's approximation to the exponential function and generalizations, arXiv:2205.08504 [math.NT], 2022.

Cf. A065973 (denominators), A005446, A005447, A264148.

# Maple program from N. J. A. Sloane, Jun 23 2011, based on J. Marsaglia's 1986 paper:
a[1]:=1;
M:=20;
for n from 2 to M do
t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k],k=2..floor(n/2));
if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;
a[n]:=t1;
od:
s1:=[seq(a[n],n=1..M)]; # This gives A005447/A005446
s2:=[seq(2^(n+1)*(n+1)!*a[2*n+2],n=0..M/2)]; # This gives A090804/A065973
# minor corrections by Peter Luschny, Nov 05 2015

Numerator@Table[(n+1)! SeriesCoefficient[-(ProductLog[-1, -Exp[-x^2-1]] + ProductLog[-Exp[-x^2-1]] + 1)/(2 x^2), {x, 0, 2n}, Assumptions -> x>0], {n, 0, 15}] (* Vladimir Reshetnikov, Nov 09 2015 *)
Numerator@Table[KroneckerDelta[n] + 2^n (3n+2)! Sum[((-1)^j 2^i StirlingS2[2n+i+j+1, j])/((2n+i+j+1)! (2n-i+1)! (i-j)! (n+i+1)), {i, 1, 2n+1}, {j, 1, i}], {n, 0, 15}]  (* Vladimir Reshetnikov, Nov 09 2015 *)

{a(n) = my(A, m); if( n<1, n==0, n++; A = vector(m=2*n, 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] * 2^n * n!))}; /* Michael Somos, Jun 09 2004 */

Define t_n by Sum_{k=0..n-1} n^k/k! + t_n*n^n/n! = exp(n)/2; then t_n ~ 1/3 + 4/(135*n) - 8/(2835*n^2) + ...
Integral_{0..infinity} exp(-x)*(1+x/n)^n dx = exp(n)*Gamma(n+1)/(2*n^n) + 2/3 - 4/(135*n) + 8/(2835*n^2) + 16/(8505*n^3) - 8992/(12629925*n^4) + ...
From Vladimir Reshetnikov, Nov 09 2015: (Start)
Numerators/denominators: A090804(n)/A065973(n) = 0^n + 2^n * (3*n+2)! / (2*n+1)! * Sum_{i=1..2*n+1} Sum_{j=1..i} Sum_{k=1..j} (-1)^k * 2^i * k^(2*n+i+j+1) * C(2*n+1,i) * C(i,j) * C(j,k) / ((2*n+i+j+1)! * (n+i+1)), where C(n,k) = A007318(n,k) are binomial coefficients.
A090804(n)/A065973(n) = 0^n + 2^n * (3*n+2)! * Sum_{i=1..2*n+1} Sum_{j=1..i} (-1)^j * 2^i * stirling2(2*n+i+j+1,j) / ((2*n+i+j+1)! * (2*n-i+1)! * (i-j)! * (n+i+1)).
(End)

a(0) corrected by Peter Luschny, Nov 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)).

A260306 Numerators in Ramanujan's asymptotic expansion of theta(n), defined by Sum_{k=0..n-1} n^k/k! + theta(n)*n^n/n! = exp(n)/2.

Original entry on oeis.org

1, 4, -8, -16, 8992, 334144, -698752, -23349012224, 1357305243136, 6319924923392, -8773495082018816, -49004477022654464, 1709650943378038784, 480380831834367035260928, -88481173388026066736939008, -660883915180095254454665216, 888962079683152174584309088256

Offset: 0

Vladimir Reshetnikov, Nov 10 2015

Let Sum_{k=0..n-1} n^k/k! + theta(n)*n^n/n! = exp(n)/2. Ramanujan gave initial terms of the asymptotic expansion theta(n) = 1/3 + (4/135)/n + (-8/2835)/n^2 + (-16/8505)/n^3 + O(1/n^4). This sequence gives the numerators in this expansion, and A065973 gives the denominators.

Ramanujan's asymptotic is also considered in the literature in the form 1-theta(n) (see for example formula (5) in the Choi link). The numerators that appear in that form are given in A090804 (the denominators are the same). The first term A090804(0) = 2 is different, and signs of other terms are opposite to a(n).

G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge, 1999; Problem 4, p. 616.
B. C. Berndt, Ramanujan's Notebooks II, Springer, 1989; p. 181, Entry 48. See also pp. 184, 193ff.
E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford Univ. Press, 1935; see p. 230, Problem 18.
S. Ramanujan, Collected Papers, edited by G. H. Hardy et al., Cambridge, 1927, pp. 323-324, Question 294.

G. C. Greubel and D. Turner, Table of n, a(n) for n = 0..117
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.
J. C. W. Marsaglia, The incomplete gamma function and Ramanujan's rational approximation to exp(x), J. Statist. Comput. Simulation, 24 (1986), 163-168.
Cormac O'Sullivan, Ramanujan's approximation to the exponential function and generalizations, arXiv:2205.08504 [math.NT], 2022.

Cf. A065973 (denominators), A090804, A264148, A005447, A005446.

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:
A260306 := n -> `if`(n=0, 1, -numer(h(2*n+1)*doublefactorial(2*n))):
seq(A260306(n), n=0..16); # Peter Luschny, Nov 20 2015

Numerator[Table[2^n*(3*n + 2)! * Sum[ Sum[ (-1)^(j + 1)*2^i*StirlingS2[2*n + i + j + 1, j]/((2*n + i + j + 1)!*(2*n - i + 1)!*(i - j)!*(n + i + 1)), {j, 1, i}], {i, 1, 2*n+1}], {n, 0, 20}]] (* Vaclav Kotesovec, Nov 20 2015 *)

Numerators/denominators: a(n)/A065973(n) = 2^n * (3*n+2)! / (2*n+1)! * Sum_{i=1..2*n+1} Sum_{j=1..i} Sum_{k=1..j} (-1)^(k+1) * 2^i * k^(2*n+i+j+1) * C(2*n+1,i) * C(i,j) * C(j,k) / ((2*n+i+j+1)! * (n+i+1)), where C(n,k) = A007318(n,k) are binomial coefficients.

a(n)/A065973(n) = 2^n * (3*n+2)! * Sum_{i=1..2*n+1} Sum_{j=1..i} (-1)^(j+1) * 2^i * stirling2(2*n+i+j+1,j) / ((2*n+i+j+1)! * (2*n-i+1)! * (i-j)! * (n+i+1)).

More terms from Vaclav Kotesovec, Nov 20 2015

A264150 Numerators of SGGS((2*n+1)/2^(n+1)) where the rational numbers SGGS(n) are defined in A264148.

Original entry on oeis.org

1, -1, 1, 1, -281, -5221, 5459, 91207079, -2650986803, -6171801683, 4283933145517, 11963983648109, -208697624924077, -29320119130515566117, 2700231121460756431181, 10084288256532215186381, -6782242429223267933535073, -51748587106835353426330148693

Offset: 0

Peter Luschny, Nov 05 2015

See A264148 for definitions and cross-references.

G. C. Greubel, Table of n, a(n) for n = 0..255

Cf. A264148, denominators in A065973.

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):
A264150 := n -> numer(SGGS(2*n+1)/2^(n+1)): seq(A264150(n), n=0..17);

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))]; b[n_] := h[n]*Factorial2[n - 1]; Table[ Numerator[b[2*n + 1]/2^(n + 1)], {n,0,50}] (* G. C. Greubel, Feb 08 2018 *)

cp's OEIS Frontend

A005446 Denominators of expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function.

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A005447 Numerators of the expansion of -W_{-1}(-e^(-1 - x^2/2)) where x > 0 and W_{-1} is the Lambert W function.

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A090804 Numerators in an asymptotic expansion of Ramanujan.

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

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A260306 Numerators in Ramanujan's asymptotic expansion of theta(n), defined by Sum_{k=0..n-1} n^k/k! + theta(n)*n^n/n! = exp(n)/2.

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A264150 Numerators of SGGS((2*n+1)/2^(n+1)) where the rational numbers SGGS(n) are defined in A264148.

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