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A274447 Numerators in expansion of W(exp(x)) about x=1, where W is the Lambert function.

%I A274447 #29 May 20 2025 21:07:55
%S A274447 1,1,1,-1,-1,13,-47,-73,2447,-16811,-15551,1726511,-18994849,10979677,
%T A274447 2983409137,-48421103257,135002366063,778870772857,-232033147779359,
%U A274447 1305952009204319,58740282660173759,-1862057132555380307,16905219421196907793,527257187244811805207
%N A274447 Numerators in expansion of W(exp(x)) about x=1, where W is the Lambert function.
%C A274447 a(17) is the first term that differs from A001662.
%H A274447 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 A274447 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 A274447 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 A274447 a(n) = A001662(n)/gcd(A001662(n),A051711(n)).
%F A274447 From _Vladimir Kruchinin_, Nov 11 2012: (Start)
%F A274447 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 A274447 a(n) = numerator((1/n!)*Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} ((-1)^j/(k-j)!)*2^(n-j-1)*Sum_{i=0..j} (1/i!)*Stirling1(n-i+j-1,j-i)/(n-i+j-1)!), n>0, a(0)=1. (End)
%F A274447 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 A274447 a(n) := numerator(1/n!*Sum_{i=1..n} Stirling2(n,i)*A013703(i)/2^(2*i+1)). - _Paolo Bonzini_, Jun 23 2016
%e A274447 W(exp(x)) = 1 + (x-1)/2 + (x-1)^2/16 - (x-1)^3/192 - ...
%p A274447 a:= n-> numer(coeftayl(LambertW(exp(x)), x=1, n)):
%p A274447 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 08 2012
%p A274447 # For large n much faster is:
%p A274447 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 A274447 seq(A001662(n), n=0..100):  # _Peter Luschny_, Nov 13 2012
%t A274447 CoefficientList[ Series[ ProductLog[ Exp[1+x] ], {x, 0, 22}], x] // Numerator (* _Jean-François Alcover_, Oct 15 2012 *)
%t A274447 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 A274447 (Sage)
%o A274447 @CachedFunction
%o A274447 def eulerian2(n, k):
%o A274447     if k==0: return 1
%o A274447     if k==n: return 0
%o A274447     return eulerian2(n-1, k)*(k+1)+eulerian2(n-1, k-1)*(2*n-k-1)
%o A274447 def q(n):
%o A274447     return add((-1)^k*eulerian2(n-1, k) for k in (0..n-1)) if n>0 else 1
%o A274447 A001662 = lambda n: (q(n)/factorial(n)).numerator()
%o A274447 [A001662(n) for n in (0..22)]  # _Peter Luschny_, Nov 13 2012
%o A274447 (Maxima)
%o A274447 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 A274447 Cf. A001662, A051711, A274448.
%K A274447 sign,easy,frac
%O A274447 0,6
%A A274447 _Paolo Bonzini_, Jun 23 2016