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%I #15 Oct 23 2020 12:50:02
%S 1,1,0,-1,-6,-78,-1544,-40605,-1328178,-51857806,-2350025232,
%T -121120896906,-6991877399100,-446673990116508,-31277285155060464,
%U -2381645560450404989,-195914136385421694954,-17312472044077536945630,-1635541992950202705979424,-164494265246550280147797438
%N G.f. A(x) satisfies: [x^(n+1)] A(x)^(n^2) = 0 for n>=0.
%F a(n) is odd iff n+1 is a power of 2 (conjecture).
%F G.f. A(x) satisfies the following relationes.
%F (1) [x^(n+1)] A(x)^(n^2) = 0 for n>=0.
%F (2) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A185072.
%F (3) A(x)/A(x)' is the g.f. of A305144. - _Paul D. Hanna_, Oct 23 2020
%e G.f.: A(x) = 1 + x - x^3 - 6*x^4 - 78*x^5 - 1544*x^6 - 40605*x^7 -...
%e Coefficients of x^k in the square powers A(x)^(n^2) of g.f. A(x) begin:
%e n=1: [1, 1, 0, -1, -6, -78, -1544, -40605, -1328178, ...];
%e n=2: [1, 4, 6, 0, -35, -396, -7182, -181824, -5817510, ...];
%e n=3: [1, 9, 36, 75, 0, -1260, -21408, -499203, -15299145, ...];
%e n=4: [1,16, 120, 544, 1484, 0, -52656, -1202240, -34269906, ...];
%e n=5: [1,25, 300, 2275, 11900, 40680, 0, -2557775, -73526475, ...];
%e n=6: [1,36, 630, 7104, 57429, 345204, 1430418, 0,-142432290, ...];
%e n=7: [1,49,1176,18375,209230,1833678,12546744, 61418175, 0, ...];
%e n=8: [1,64,2016,41600,630960,7470336,71271616,549420288,3113335320, 0, ...]; ...
%e where the coefficients of x^(n+1) in A(x)^(n^2) all equal zero for n>=0.
%e Related expansions.
%e A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A185072:
%e G(x) = 1 + x - 2*x^2 + 6*x^3 - 28*x^4 + 70*x^5 - 1446*x^6 -...
%e A(x)'/A(x) = 1 - x - 2*x^2 - 21*x^3 - 364*x^4 - 8830*x^5 - 273972*x^6 - 10313037*x^7 - 455135384*x^8 - 22995056286*x^9 - 1307053358940*x^10 - ...
%e A(x)/A(x)' = 1 + x + 3*x^2 + 26*x^3 + 417*x^4 + 9726*x^5 + 295000*x^6 + 10946172*x^7 + 478392123*x^8 + ... + A305144(n)*x^n + ...
%o (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^2),k+1));A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A185072, A305144, A230218, A229041, A171791.
%K sign
%O 0,5
%A _Paul D. Hanna_, Sep 12 2013