cp's OEIS Frontend

A352412 E.g.f.: 2*x / LambertW( 2*x/(1-x) ).

%I A352412 #15 Mar 17 2022 14:54:30
%S A352412 1,1,-4,20,-224,3392,-67232,1629728,-46799104,1552143104,-58386807296,
%T A352412 2455954797056,-114222622662656,5819845970653184,-322384671892123648,
%U A352412 19290013218140254208,-1239886482366130946048,85200320552417960394752
%N A352412 E.g.f.: 2*x / LambertW( 2*x/(1-x) ).
%C A352412 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 A352412 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
%F A352412 (1) A(x) = 2*x / LambertW( 2*x/(1-x) ).
%F A352412 (2) A(x) = (1-x) * exp( 2*x/A(x) ).
%F A352412 (3) A(x) = 2*x / log( A(x)/(1-x) ).
%F A352412 (4) A( x/(exp(-2*x) + x) ) = 1/(exp(-2*x) + x).
%F A352412 (5) A(x) = x / Series_Reversion( x/(exp(-2*x) + x) ).
%F A352412 (6) Sum_{k=0..n} [x^k] A(x)^n = 0, for n > 1.
%F A352412 (7) [x^(n+1)/(n+1)!] A(x)^n = -(-2)^(n+1) * n for n >= (-1).
%F A352412 a(n) ~ (-1)^(n+1) * exp(-1) * sqrt(2) * (2 - exp(-1))^(n - 1/2) * n^(n-1). - _Vaclav Kotesovec_, Mar 15 2022
%e A352412 E.g.f.: A(x) = 1 + x - 4*x^2/2! + 20*x^3/3! - 224*x^4/4! + 3392*x^5/5! - 67232*x^6/6! + 1629728*x^7/7! - 46799104*x^8/8! + ...
%e A352412 such that A(x) = (1-x) * exp(2*x/A(x)), where
%e A352412 exp(2*x/A(x)) = 1 + 2*x + 20*x^3/3! - 144*x^4/4! + 2672*x^5/5! - 51200*x^6/6! + 1271328*x^7/7! - 36628480*x^8/8! + ...
%e A352412 Related series.
%e A352412 The e.g.f. A(x) satisfies A( x/(exp(-2*x) + x) ) = 1/(exp(-2*x) + x), where
%e A352412 1/(exp(-2*x) + x) = 1 + x - 2*x^2/2! - 10*x^3/3! + 24*x^4/4! + 312*x^5/5! - 560*x^6/6! + ... + A336958(n)*(-x)^n/n! + ...
%e A352412 Related table.
%e A352412 Another defining property of the e.g.f. A(x) is illustrated here.
%e A352412 The table of coefficients of x^k/k! in A(x)^n begins:
%e A352412 n=1: [1, 1, -4,   20,  -224, 3392, -67232, 1629728, ...];
%e A352412 n=2: [1, 2, -6,   16,  -192, 2944, -58880, 1434752, ...];
%e A352412 n=3: [1, 3, -6,   -6,   -48, 1296, -29664,  776544, ...];
%e A352412 n=4: [1, 4, -4,  -40,    88,  128,  -7424,  263936, ...];
%e A352412 n=5: [1, 5,  0,  -80,   120,  280,   -320,   38720, ...];
%e A352412 n=6: [1, 6,  6, -120,   -24, 1872,  -3312,     768, ...];
%e A352412 n=7: [1, 7, 14, -154,  -392, 4424,  -3920,  -22288, ...];
%e A352412 ...
%e A352412 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 A352412 n=1: 2 = 1 + 1;
%e A352412 n=2: 0 = 1 + 2 + -6/2!;
%e A352412 n=3: 0 = 1 + 3 + -6/2! +   -6/3!;
%e A352412 n=4: 0 = 1 + 4 + -4/2! +  -40/3! +   88/4!;
%e A352412 n=5: 0 = 1 + 5 +  0/2! +  -80/3! +  120/4! +  280/5!;
%e A352412 n=6: 0 = 1 + 6 +  6/2! + -120/3! +  -24/4! + 1872/5! + -3312/6!;
%e A352412 n=7: 0 = 1 + 7 + 14/2! + -154/3! + -392/4! + 4424/5! + -3920/6! + -22288/7!;
%e A352412 ...
%o A352412 (PARI) {a(n) = n!*polcoeff( x/serreverse( x/(exp(-2*x  +x^2*O(x^n)) + x) ),n)}
%o A352412 for(n=0,30,print1(a(n),", "))
%o A352412 (PARI) my(x='x+O('x^30)); Vec(serlaplace(2*x/lambertw(2*x/(1-x)))) \\ _Michel Marcus_, Mar 17 2022
%Y A352412 Cf. A352410, A352411, A352448, A336958.
%K A352412 sign
%O A352412 0,3
%A A352412 _Paul D. Hanna_, Mar 15 2022