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A229041 G.f. A(x) satisfies: [x^n] A(x)^(n^2-n+1) = 0 for n>=2.

%I A229041 #12 Apr 20 2014 22:29:15
%S A229041 1,1,-1,1,-7,-49,-1191,-31569,-1051695,-41520593,-1896894223,
%T A229041 -98362962257,-5705059841823,-365846227736001,-25696840682622175,
%U A229041 -1961769357361345473,-161728572333727674687,-14318505129615014956737,-1354916705432679538845759,-136467389971873491004759617
%N A229041 G.f. A(x) satisfies: [x^n] A(x)^(n^2-n+1) = 0 for n>=2.
%C A229041 G.f. A(x) satisfies:
%C A229041 (1) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A185072.
%C A229041 (2) A(x) = F(x/A(x)) where F(x) = A(x*F(x)) is the g.f. of A229044.
%e A229041 G.f.: A(x) = 1 + x - x^2 + x^3 - 7*x^4 - 49*x^5 - 1191*x^6 - 31569*x^7 -...
%e A229041 Coefficients of x^k in the powers A(x)^(n^2-n+1) of g.f. A(x) begin:
%e A229041 n=1: [1, 1,  -1,    1,    -7,    -49,   -1191,   -31569,  -1051695, ...];
%e A229041 n=2: [1, 3,   0,   -2,   -15,   -189,   -3850,  -101700,  -3340845, ...];
%e A229041 n=3: [1, 7,  14,    0,   -56,   -588,  -10808,  -273972,  -8760325, ...];
%e A229041 n=4: [1,13,  65,  143,     0,  -1742,  -27534,  -638690, -19496334, ...];
%e A229041 n=5: [1,21, 189,  931,  2478,      0,  -67312, -1444608, -40653711, ...];
%e A229041 n=6: [1,31, 434, 3596, 19158,  62062,       0, -3116120, -84939504, ...];
%e A229041 n=7: [1,43, 860,10578, 88795, 526449, 2045854,        0,-167991196, ...];
%e A229041 n=8: [1,57,1539,26125,311619,2754297,18283187, 83718693,         0, ...];
%e A229041 ...
%e A229041 where the coefficients of x^n in A(x)^(n^2-n+1) all equal zero for n>=2.
%o A229041 (PARI) {a(n)=local(A=[1,1]);for(k=1,n,A=concat(A,0);A[#A]=-polcoeff((Ser(A) +O(x^(k+2)))^(k^2+k+1)/(k^2+k+1),k+1));A[n+1]}
%o A229041 for(n=0,30,print1(a(n),", "))
%Y A229041 Cf. A185072, A229044, A230218, A171791.
%K A229041 sign
%O A229041 0,5
%A A229041 _Paul D. Hanna_, Sep 14 2013