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%I A350525 #5 Jan 07 2022 12:01:08
%S A350525 1,6,-36,360,-4320,57024,-816480,12737088,-216950400,3985853184,
%T A350525 -77049957888,1524488905728,-30107409053184,576921511037952,
%U A350525 -9954313034508288,94929987120648192,5615966537573179392,-686220690340602003456,57421269146260788903936
%N A350525 G.f. A(x) satisfies: [x^(2*n-2)] A(x)^(n*(n+1)/2) = 0 and [x^(2*n-1)] A(x)^(n*(n+1)/2) = 0 for n > 1, with a(0) = 1, a(2) = 6.
%e A350525 G.f.: A(x) = 1 + 6*x - 36*x^2 + 360*x^3 - 4320*x^4 + 57024*x^5 - 816480*x^6 + 12737088*x^7 - 216950400*x^8 + 3985853184*x^9 + ...
%e A350525 The table of coefficients of x^k in A(x)^(n*(n+1)/2), for k>=0, begins:
%e A350525 n=1: [1, 6, -36, 360, -4320, 57024, -816480, 12737088, -216950400, ...];
%e A350525 n=2: [1, 18, 0, 0, 0, 0, -54432, 2659392, -85660416, 2267481600, -...];
%e A350525 n=3: [1, 36, 324, 0, 0, 0, -108864, 3359232, -75582720, 1451188224, ...];
%e A350525 n=4: [1, 60, 1260, 10080, 15120, -36288, 0, 0, 5598720, -245099520, ...];
%e A350525 n=5: [1, 90, 3240, 58320, 524880, 1889568, -272160, -6298560, 0, 0, 0, 0, ...]; ...
%e A350525 in which both coefficients of x^(2*n-2) and x^(2*n-1) in A(x)^(n^2) equal zero for n > 1.
%o A350525 (PARI) {a(n) = my(A=[1,6],P); for(i=1,n, A=concat(A,0); P = (#A+1)\2;
%o A350525 A[#A] = -polcoeff( Ser(A)^(P*(P+1)/2)/(P*(P+1)/2), #A-1) ); H=A; A[n+1]}
%o A350525 for(n=0,20, print1(a(n),", "))
%Y A350525 Cf. A250117, A350524.
%K A350525 sign
%O A350525 0,2
%A A350525 _Paul D. Hanna_, Jan 03 2022

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