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A230218 G.f. A(x) satisfies: [x^n] A(x)^(n^2+n+1) = 0 for n>1.

%I A230218 #18 Sep 27 2017 04:39:28
%S A230218 1,1,-3,14,-85,504,-4424,6796,-878157,-25703710,-1270518018,
%T A230218 -65772588300,-3848787714746,-248212765567326,-17520121174143210,
%U A230218 -1343050785659060872,-111112550557260635229,-9867409274482580015370,-936234289413196544207234,-94522404087905722536648780
%N A230218 G.f. A(x) satisfies: [x^n] A(x)^(n^2+n+1) = 0 for n>1.
%H A230218 Paul D. Hanna, <a href="/A230218/b230218.txt">Table of n, a(n) for n = 0..520</a>
%F A230218 for n>0, a(n) is odd iff n is a power of 2 (conjecture).
%F A230218 G.f. A(x) satisfies:
%F A230218 (1) A(x) = F(x/A(x)) where F(x) = A(x*F(x)) is the g.f. of A185072.
%F A230218 (2) A(x) = G(x/A(x)^2) where G(x) = A(x*G(x)^2) is the g.f. of A229041.
%F A230218 a(n) ~ -c * 2^(2*n) *n^(n-5/2) / (exp(n) * d^n * (2-d)^n), where d = -LambertW(-2*exp(-2)) = -A226775 = 0.40637573995995990767695812412483975821... and c = 0.015106126717978... - _Vaclav Kotesovec_, Sep 27 2017
%e A230218 G.f.: A(x) = 1 + x - 3*x^2 + 14*x^3 - 85*x^4 + 504*x^5 - 4424*x^6 +...
%e A230218 Coefficients of x^k in the powers A(x)^(n^2+n+1) of g.f. A(x) begin:
%e A230218 n=0: [1, 1,   -3,    14,    -85,     504,    -4424,      6796, ...];
%e A230218 n=1: [1, 3,   -6,    25,   -153,     819,    -8664,    -18360, ...];
%e A230218 n=2: [1, 7,    0,     7,    -98,     210,   -10122,   -141525, ...];
%e A230218 n=3: [1,13,   39,     0,    -78,    -819,   -15483,   -380952, ...];
%e A230218 n=4: [1,21,  147,   364,      0,   -2457,   -35805,   -821916, ...];
%e A230218 n=5: [1,31,  372,  2139,   5580,       0,   -91698,  -1792947, ...];
%e A230218 n=6: [1,43,  774,  7525,  42097,  125517,        0,  -4097298, ...];
%e A230218 n=7: [1,57, 1425, 20482, 185877, 1089270,  3791298,         0, ...];
%e A230218 n=8: [1,73, 2409, 47450, 619697, 5619978, 35621518, 144591976, 0, ...]; ...
%e A230218 where the coefficients of x^n in A(x)^(n^2+n+1) all equal zero for n>1.
%e A230218 ODD TERMS:
%e A230218 For n>0, a(n) appears to be odd only when n is a power of 2:
%e A230218 a(1) = 1;
%e A230218 a(2) = -3;
%e A230218 a(4) = -85;
%e A230218 a(8) = -878157;
%e A230218 a(16) = -111112550557260635229;
%e A230218 a(32) = -886203693344229341179357569730608605545213045330679133; ...
%o A230218 (PARI) {a(n)=local(A=[1,1]); for(m=2,n, A=concat(A, 0); A[#A]=-Vec(Ser(A)^(m^2+m+1))[m+1]/(m^2+m+1)); A[n+1]}
%o A230218 for(n=0,20,print1(a(n),", "))
%Y A230218 Cf. A185072, A229041, A229044, A171791, A292877.
%K A230218 sign
%O A230218 0,3
%A A230218 _Paul D. Hanna_, Oct 11 2013