A350525 - OEIS

%I #5 Jan 07 2022 12:01:08

%S 1,6,-36,360,-4320,57024,-816480,12737088,-216950400,3985853184,

%T -77049957888,1524488905728,-30107409053184,576921511037952,

%U -9954313034508288,94929987120648192,5615966537573179392,-686220690340602003456,57421269146260788903936

%N 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 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 The table of coefficients of x^k in A(x)^(n*(n+1)/2), for k>=0, begins:

%e n=1: [1, 6, -36, 360, -4320, 57024, -816480, 12737088, -216950400, ...];

%e n=2: [1, 18, 0, 0, 0, 0, -54432, 2659392, -85660416, 2267481600, -...];

%e n=3: [1, 36, 324, 0, 0, 0, -108864, 3359232, -75582720, 1451188224, ...];

%e n=4: [1, 60, 1260, 10080, 15120, -36288, 0, 0, 5598720, -245099520, ...];

%e n=5: [1, 90, 3240, 58320, 524880, 1889568, -272160, -6298560, 0, 0, 0, 0, ...]; ...

%e 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 (PARI) {a(n) = my(A=[1,6],P); for(i=1,n, A=concat(A,0); P = (#A+1)\2;

%o A[#A] = -polcoeff( Ser(A)^(P*(P+1)/2)/(P*(P+1)/2), #A-1) ); H=A; A[n+1]}

%o for(n=0,20, print1(a(n),", "))

%Y Cf. A250117, A350524.

%K sign

%O 0,2

%A _Paul D. Hanna_, Jan 03 2022