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%I #11 Mar 11 2018 20:09:40
%S 1,1,4,51,1176,40160,1852788,109746196,8076254064,720996043500,
%T 76639917914560,9553107629947662,1378566504590365848,
%U 227792196676700222749,42692948669436573948412,9000830359064469548726190,2119119760287990864773870816,553573708504386843864377357273,159532387646232647024272055366172,50459353730921637796726256001307227
%N G.f. A(x) satisfies: [x^n] A( x/A(x)^(2*n) ) = 0 for n>=1.
%C Conjecture: n divides a(n) for n>=1.
%H Paul D. Hanna, <a href="/A300732/b300732.txt">Table of n, a(n) for n = 0..300</a>
%e G.f.: A(x) = 1 + x + 4*x^2 + 51*x^3 + 1176*x^4 + 40160*x^5 + 1852788*x^6 + 109746196*x^7 + 8076254064*x^8 + 720996043500*x^9 + 76639917914560*x^10 + ...
%e The table of coefficients in A( x/A(x)^(2*n) ) begins:
%e n=0: [1, 1, 4, 51, 1176, 40160, 1852788, 109746196, 8076254064, ...];
%e n=1: [1, 1, 2, 30, 764, 27982, 1358312, 83578886, 6332601998, ...];
%e n=2: [1, 1, 0, 13, 436, 18185, 954672, 61915953, 4870913328, ...];
%e n=3: [1, 1, -2, 0, 184, 10453, 628566, 44079848, 3649012438, ...];
%e n=4: [1, 1, -4, -9, 0, 4486, 367876, 29468486, 2629920072, ...];
%e n=5: [1, 1, -6, -14, -124, 0, 161636, 17550750, 1781417478, ...];
%e n=6: [1, 1, -8, -15, -196, -3273, 0, 7862059, 1075627896, ...];
%e n=7: [1, 1, -10, -12, -224, -5585, -125790, 0, 488615262, ...];
%e n=8: [1, 1, -12, -5, -216, -7172, -223436, -6379976, 0, ...];
%e n=9: [1, 1, -14, 6, -180, -8254, -299616, -11568794, -407408226, 0, ...]; ...
%e such that the main diagonal consists of all zeros after the initial terms.
%e The terms a(n)/n for n>=1 begin:
%e [1, 2, 17, 294, 8032, 308798, 15678028, 1009531758, 80110671500, ...].
%o (PARI) {a(n) = my(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[#A] = -Vec(subst(Ser(A), x, x/Ser(A)^(2*(#A-1))))[#A]); A[n+1]}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A266489, A300733.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 11 2018
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