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Numerators in expansion of W(exp(x)) about x=1, where W is the Lambert function.
3

%I #26 Mar 17 2018 18:28:04

%S 1,1,1,-1,-1,13,-47,-73,2447,-16811,-15551,1726511,-18994849,10979677,

%T 2983409137,-48421103257,135002366063,778870772857,-232033147779359,

%U 1305952009204319,58740282660173759,-1862057132555380307,16905219421196907793,527257187244811805207

%N Numerators in expansion of W(exp(x)) about x=1, where W is the Lambert function.

%C a(17) is the first term that differs from A001662.

%H Alois P. Heinz, <a href="/A274447/b274447.txt">Table of n, a(n) for n = 0..446</a> (first 151 terms from G. C. Greubel)

%H R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, <a href="http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/LambertW.ps">On the Lambert W Function</a>, Advances in Computational Mathematics, (5), 1996, pp. 329-359.

%H R. M. Corless, D. J. Jeffrey and D. E. Knuth, <a href="http://www.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/CorlessJeffreyKnuth.ps">A sequence of series for the Lambert W Function</a> (section 2.2).

%F a(n) = A001662(n)/gcd(A001662(n),A051711(n)).

%F From _Vladimir Kruchinin_, Nov 11 2012: (Start)

%F a(n) = numerator(1/n!*(Sum_{u=2..n} stirling2(n,u)*(Sum_{k=1..u-1} ((u+k-1)!*Sum_{j=1..k} 2^(-u-j)/(k-j)!*Sum_{l=1..j} (-1)^(l)/((j-l)!)*Sum_{i=0..l} (l^(u+j-i-1))/((l-i)!*i!*(u+j-i-1)!)))+1/2)).

%F a(n) = numerator(1/n!*Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} ((-1)^(j)/(k-j)!*Sum_{i=0..j} ((1/i!)*Stirling1(n-i+j-1,j-i))/(n-i+j-1)!))*2^(n-j-1)), n>0, a(0)=1. (End)

%F a(n) = numerator(q(n)/n!) where q(n) = add_{k=0..n-1}(-1)^k*E2(n-1,k) if n>0 and 1 otherwise, E2 the second-order Eulerian numbers. - _Peter Luschny_, Nov 13 2012

%F a(n) := numerator(1/n!*Sum_{i=1..n} Stirling2(n,i)*A013703(i)/2^(2*i+1)). - _Paolo Bonzini_, Jun 23 2016

%e W(exp(x)) = 1 + (x-1)/2 + (x-1)^2/16 - (x-1)^3/192 - ...

%p a:= n-> numer(coeftayl(LambertW(exp(x)), x=1, n)):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Nov 08 2012

%p # For large n much faster is:

%p q := proc(n) if n=0 then 1 else add((-1)^k*combinat[eulerian2](n-1, k), k=0..n-1) fi end: A001662 := n -> numer(q(n)/n!):

%p seq(A001662(n), n=0..100): # _Peter Luschny_, Nov 13 2012

%t CoefficientList[ Series[ ProductLog[ Exp[1+x] ], {x, 0, 22}], x] // Numerator (* _Jean-François Alcover_, Oct 15 2012 *)

%t a[0] = 1; a[n_] := 1/n!*Sum[(n+k-1)!*Sum[(-1)^(j)/(k-j)!*Sum[1/i!* StirlingS1[n-i+j-1, j-i]/(n-i+j-1)!, {i, 0, j}]*2^(n-j-1), {j, 0, k}], {k, 0, n-1}] // Numerator; Array[a, 30, 0] (* _Jean-François Alcover_, Feb 13 2016, after _Vladimir Kruchinin_ *)

%o (Sage)

%o @CachedFunction

%o def eulerian2(n, k):

%o if k==0: return 1

%o if k==n: return 0

%o return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)

%o def q(n):

%o return add((-1)^k*eulerian2(n-1, k) for k in (0..n-1)) if n>0 else 1

%o A001662 = lambda n: (q(n)/factorial(n)).numerator()

%o [A001662(n) for n in (0..22)] # _Peter Luschny_, Nov 13 2012

%o (Maxima)

%o a(n):=num(if n=0 then 1 else 1/n!*(sum((n+k-1)!*sum(((-1)^(j)/(k-j)!*sum((1/i!*stirling1(n-i+j-1, j-i))/(n-i+j-1)!, i, 0, j))*2^(n-j-1), j, 0, k), k, 0, n-1))); /* _Vladimir Kruchinin_, Nov 11 2012 */

%Y Cf. A001662, A051711, A274448.

%K sign,easy,frac

%O 0,6

%A _Paolo Bonzini_, Jun 23 2016