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A352410 Expansion of e.g.f. LambertW( -x/(1-x) ) / (-x).

%I A352410 #38 Aug 24 2025 11:00:08
%S A352410 1,2,9,67,717,10141,179353,3816989,95076537,2714895433,87457961421,
%T A352410 3138260371225,124147801973605,5368353187693757,251928853285058433,
%U A352410 12752446755011776741,692625349011401620209,40178978855796929378065,2479383850197948228950293
%N A352410 Expansion of e.g.f. LambertW( -x/(1-x) ) / (-x).
%C A352410 An interesting property of this e.g.f. A(x) is that the sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1.
%H A352410 Seiichi Manyama, <a href="/A352410/b352410.txt">Table of n, a(n) for n = 0..370</a>
%F A352410 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
%F A352410 (1) A(x) = LambertW( -x/(1-x) ) / (-x).
%F A352410 (2) A(x) = exp( x*A(x) ) / (1-x).
%F A352410 (3) A(x) = log( (1-x) * A(x) ) / x.
%F A352410 (4) A( x/(exp(x) + x) ) = exp(x) + x.
%F A352410 (5) A(x) = (1/x) * Series_Reversion( x/(exp(x) + x) ).
%F A352410 (6) Sum_{k=0..n} [x^k] 1/A(x)^n = 0, for n > 1.
%F A352410 (7) [x^(n+1)/(n+1)!] 1/A(x)^n = -n for n >= (-1).
%F A352410 a(n) ~ (1 + exp(1))^(n + 3/2) * n^(n-1) / exp(n + 1/2). - _Vaclav Kotesovec_, Mar 15 2022
%F A352410 a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n,k)/k!. - _Seiichi Manyama_, Sep 24 2022
%e A352410 E.g.f.: A(x) = 1 + 2*x + 9*x^2/2! + 67*x^3/3! + 717*x^4/4! + 10141*x^5/5! + 179353*x^6/6! + 3816989*x^7/7! + ...
%e A352410 such that A(x) = exp(x*A(x)) / (1-x), where
%e A352410 exp(x*A(x)) = 1 + x + 5*x^2/2! + 40*x^3/3! + 449*x^4/4! + 6556*x^5/5! + 118507*x^6/6! + ... + A052868(n)*x^n/n! + ...
%e A352410 which equals LambertW(-x/(1-x)) * (1-x)/(-x).
%e A352410 Related table.
%e A352410 Another defining property of the e.g.f. A(x) is illustrated here.
%e A352410 The table of coefficients of x^k/k! in 1/A(x)^n begins:
%e A352410 n=1: [1,  -2,  -1,    -7,   -71,   -961, -16409, -339571, ...];
%e A352410 n=2: [1,  -4,   6,    -2,   -24,   -362,  -6644, -144538, ...];
%e A352410 n=3: [1,  -6,  21,   -33,    -3,    -63,  -1395,  -34275, ...];
%e A352410 n=4: [1,  -8,  44,  -148,   232,     -4,   -152,   -4876, ...];
%e A352410 n=5: [1, -10,  75,  -395,  1305,  -2045,     -5,    -355, ...];
%e A352410 n=6: [1, -12, 114,  -822,  4224, -13806,  21636,      -6, ...];
%e A352410 n=7: [1, -14, 161, -1477, 10381, -52507, 170401, -267043, -7, ...];
%e A352410 ...
%e A352410 from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1, as follows:
%e A352410 n=1:-1 = 1 +  -2;
%e A352410 n=2: 0 = 1 +  -4 +   6/2!;
%e A352410 n=3: 0 = 1 +  -6 +  21/2! +   -33/3!;
%e A352410 n=4: 0 = 1 +  -8 +  44/2! +  -148/3! +   232/4!;
%e A352410 n=5: 0 = 1 + -10 +  75/2! +  -395/3! +  1305/4! +  -2045/5!;
%e A352410 n=6: 0 = 1 + -12 + 114/2! +  -822/3! +  4224/4! + -13806/5! +  21636/6!;
%e A352410 n=7: 0 = 1 + -14 + 161/2! + -1477/3! + 10381/4! + -52507/5! + 170401/6! + -267043/7!;
%e A352410 ...
%t A352410 terms = 19; A[_] = 0; Do[A[x_] = Exp[x*A[x]]/(1-x) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* _Stefano Spezia_, Mar 24 2025 *)
%t A352410 With[{nn=20},CoefficientList[Series[LambertW[-x/(1-x)]/-x,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 24 2025 *)
%o A352410 (PARI) {a(n) = n!*polcoeff( (1/x)*serreverse( x/(exp(x +x^2*O(x^n)) + x) ),n)}
%o A352410 for(n=0,30,print1(a(n),", "))
%o A352410 (PARI) my(x='x+O('x^30)); Vec(serlaplace(lambertw(-x/(1-x))/(-x))) \\ _Michel Marcus_, Mar 17 2022
%o A352410 (PARI) a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n, k)/k!); \\ _Seiichi Manyama_, Sep 24 2022
%Y A352410 Cf. A352411, A352412, A352448, A052868.
%Y A352410 Cf. A102743, A108919, A331726.
%K A352410 nonn,changed
%O A352410 0,2
%A A352410 _Paul D. Hanna_, Mar 15 2022