A005446 Denominators of expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function. (Formerly M3140)

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

Offset

0,3

Comments

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.

References

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).

Links

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]

Formula

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

Example

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 + ...

Maple

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)];

Mathematica

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 *)

Prog

(PARI) 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 */
(PARI) 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 */
(SageMath)
@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

Crossrefs

Cf. A005447, A090804/A065973.

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

Sequence in context: A073992 A268898 A127960 * A056307 A056299 A212616

Adjacent sequences: A005443 A005444 A005445 * A005447 A005448 A005449

Keyword

nonn,frac

Author

N. J. A. Sloane

Extensions

Edited by Michael Somos, Jul 21 2002

Status

approved