This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A352411 #15 Mar 17 2022 14:54:18
%S A352411 1,0,-1,1,-7,31,-281,2381,-28015,346879,-5149009,82769149,-1499707991,
%T A352411 29444151023,-632715633577,14631547277101,-364321853163871,
%U A352411 9686058045625471,-274387229080161569,8241211775883617405,-261766195805536280839,8763341168691985628719
%N A352411 E.g.f.: x / LambertW( x/(1-x) ).
%C A352411 An interesting property of this e.g.f. A(x) is that the sum of coefficients of x^k, k=0..n, in A(x)^n equals zero, for n > 1.
%F A352411 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
%F A352411 (1) A(x) = x / LambertW( x/(1-x) ).
%F A352411 (2) A(x) = (1-x) * exp( x/A(x) ).
%F A352411 (3) A(x) = x / log( A(x)/(1-x) ).
%F A352411 (4) A( x/(exp(-x) + x) ) = 1/(exp(-x) + x).
%F A352411 (5) A(x) = x / Series_Reversion( x/(exp(-x) + x) ).
%F A352411 (6) Sum_{k=0..n} [x^k] A(x)^n = 0, for n > 1.
%F A352411 (7) [x^(n+1)/(n+1)!] A(x)^n = (-1)^n * n for n >= (-1).
%F A352411 a(n) ~ (-1)^(n+1) * exp(-1) * (1 - exp(-1))^(n - 1/2) * n^(n-1). - _Vaclav Kotesovec_, Mar 15 2022
%e A352411 E.g.f.: A(x) = 1 + 0*x - x^2/2! + x^3/3! - 7*x^4/4! + 31*x^5/5! - 281*x^6/6! + 2381*x^7/7! - 28015*x^8/8! + ...
%e A352411 such that A(x) = (1-x) * exp(x/A(x)), where
%e A352411 exp(x/A(x)) = 1 + x + x^2/2! + 4*x^3/3! + 9*x^4/4! + 76*x^5/5! + 175*x^6/6! + 3606*x^7/7! + 833*x^8/8! + ...
%e A352411 Related series.
%e A352411 The e.g.f. A(x) satisfies A( x/(exp(-x) + x) ) = 1/(exp(-x) + x), where
%e A352411 1/(exp(-x) + x) = 1 - x^2/2! + x^3/3! + 5*x^4/4! - 19*x^5/5! - 41*x^6/6! + 519*x^7/7! - 183*x^8/8! + ...
%e A352411 Related table.
%e A352411 Another defining property of the e.g.f. A(x) is illustrated here.
%e A352411 The table of coefficients of x^k/k! in A(x)^n begins:
%e A352411 n=1: [1, 0, -1, 1, -7, 31, -281, 2381, -28015, ...];
%e A352411 n=2: [1, 0, -2, 2, -8, 42, -332, 2970, -33392, ...];
%e A352411 n=3: [1, 0, -3, 3, -3, 33, -243, 2397, -26631, ...];
%e A352411 n=4: [1, 0, -4, 4, 8, 4, -104, 1292, -15712, ...];
%e A352411 n=5: [1, 0, -5, 5, 25, -45, -5, 285, -6095, ...];
%e A352411 n=6: [1, 0, -6, 6, 48, -114, -36, 6, -720, ...];
%e A352411 n=7: [1, 0, -7, 7, 77, -203, -287, 1085, -7, ...];
%e A352411 n=8: [1, 0, -8, 8, 112, -312, -848, 4152, -1856, 8, ...];
%e A352411 ...
%e A352411 from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in A(x)^n equals zero, for n > 1, as follows:
%e A352411 n=1: 1 = 1 + 0;
%e A352411 n=2: 0 = 1 + 0 + -2/2!;
%e A352411 n=3: 0 = 1 + 0 + -3/2! + 3/3!;
%e A352411 n=4: 0 = 1 + 0 + -4/2! + 4/3! + 8/4!;
%e A352411 n=5: 0 = 1 + 0 + -5/2! + 5/3! + 25/4! + -45/5!;
%e A352411 n=6: 0 = 1 + 0 + -6/2! + 6/3! + 48/4! + -114/5! + -36/6!;
%e A352411 n=7: 0 = 1 + 0 + -7/2! + 7/3! + 77/4! + -203/5! + -287/6! + 1085/7!;
%e A352411 n=8: 0 = 1 + 0 + -8/2! + 8/3! + 112/4! + -312/5! + -848/6! + 4152/7! + -1856/8!;
%e A352411 ...
%o A352411 (PARI) {a(n) = n!*polcoeff( x/serreverse( x/(exp(-x +x^2*O(x^n)) + x) ),n)}
%o A352411 for(n=0,30,print1(a(n),", "))
%o A352411 (PARI) my(x='x+O('x^30)); Vec(serlaplace(x/lambertw(x/(1-x)))) \\ _Michel Marcus_, Mar 17 2022
%Y A352411 Cf. A352410, A352412, A352448.
%K A352411 sign
%O A352411 0,5
%A A352411 _Paul D. Hanna_, Mar 15 2022