A350524 - OEIS

%I #6 Jan 07 2022 12:01:44

%S 1,2,-6,28,-144,736,-3512,14896,-61600,509632,-12903296,422568704,

%T -17796848640,824388274176,-43343785743488,2375499099860224,

%U -140774447935008256,8727751762659943424,-578377033389467758592,40230774454685666598912,-2967831511563656631672832,229648123356288830870929408

%N G.f. A(x) satisfies: [x^(2*n-2)] A(x)^(n^2) = 0 and [x^(2*n-1)] A(x)^(n^2) = 0 for n > 1, with a(0) = 1, a(2) = 2.

%e G.f.: A(x) = 1 + 2*x - 6*x^2 + 28*x^3 - 144*x^4 + 736*x^5 - 3512*x^6 + 14896*x^7 - 61600*x^8 + 509632*x^9 - 12903296*x^10 + ...

%e The table of coefficients of x^k in A(x)^(n^2), for k>=0, begins:

%e n=1: [1, 2, -6, 28, -144, 736, -3512, 14896, -61600, 509632, ...];

%e n=2: [1, 8, 0, 0, 40, -512, 4608, -32768, 152272, 456064, ...];

%e n=3: [1, 18, 90, 60, 0, 0, -1176, 20592, -278208, 3442304, ...];

%e n=4: [1, 32, 384, 2048, 4256, 1792, 0, 0, -36672, 621568, ...];

%e n=5: [1, 50, 1050, 11900, 77600, 285760, 537000, 399600, 0, 0, ...];

%e n=6: [1, 72, 2304, 43008, 516456, 4147200, 22411776, 79921152, 178965072, 227782016, 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=[1,2],P); for(i=1,n, A=concat(A,0); P = (#A+1)\2;

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

%o for(n=0,30, print1(a(n),", "))

%Y Cf. A171791, A350525.

%K sign

%O 0,2

%A _Paul D. Hanna_, Jan 03 2022