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G.f. A(x) satisfies: [x^n] A( x/A(x)^(n^2) ) = 0 for n>1.
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%I #9 Mar 11 2018 20:25:02

%S 1,1,4,63,2160,138025,15073596,2572772048,642195593408,

%T 224210211292629,105538947969000700,64968095569659693840,

%U 51052911516803277805536,50200764753583055762144696,60726306448248686394356127312,89059256525528149546284788980800,156362202771639017446505434119944960,325036812372664804706525309628909742683

%N G.f. A(x) satisfies: [x^n] A( x/A(x)^(n^2) ) = 0 for n>1.

%C a(n) is divisible by n^2 for n>=1 (conjecture): A292395(n) = a(n)/n^2.

%H Paul D. Hanna, <a href="/A292394/b292394.txt">Table of n, a(n) for n = 0..200</a>

%e G.f.: A(x) = 1 + x + 4*x^2 + 63*x^3 + 2160*x^4 + 138025*x^5 + 15073596*x^6 + 2572772048*x^7 + 642195593408*x^8 + 224210211292629*x^9 + 105538947969000700*x^10 +...

%e The table of coefficients in A(x/A(x)^(n^2)) begins:

%e n=0: [1, 1, 4, 63, 2160, 138025, 15073596, 2572772048, 642195593408, ...];

%e n=1: [1, 1, 3, 52, 1895, 126554, 14211827, 2465030616, 621409924326, ...];

%e n=2: [1, 1, 0, 25, 1228, 96586, 11876160, 2164945551, 562519626368, ...];

%e n=3: [1, 1, -5, 0, 483, 59116, 8696349, 1732963772, 474868166564, ...];

%e n=4: [1, 1, -12, 7, 0, 25509, 5385244, 1244512060, 370970521936, ...];

%e n=5: [1, 1, -21, 88, -165, 0, 2430241, 766004308, 263319541888, ...];

%e n=6: [1, 1, -32, 297, -660, -24926, 0, 342132167, 161945375456, ...];

%e n=7: [1, 1, -45, 700, -3377, -64394, -1721889, 0, 73311075962, ...];

%e n=8: [1, 1, -60, 1375, -12112, -120679, -1685572, -241494912, 0, ...];

%e n=9: [1, 1, -77, 2412, -33345, -137186, 3227751, -394368976, -60181947214, 0, ...]; ...

%e such that the main diagonal consists of all zeros after the initial terms.

%e The values for a(n)/n, for n>=1, begin:

%e [1, 2, 21, 540, 27605, 2512266, 367538864, 80274449176, 24912245699181, 10553894796900070, 5906190506332699440, ...].

%e The values for a(n)/n^2, for n>=1, form A292395 and begin:

%e [1, 1, 7, 135, 5521, 418711, 52505552, 10034306147, 2768027299909, 1055389479690007, 536926409666609040, ...].

%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)^((#A-1)^2)))[#A]); A[n+1]}

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

%Y Cf. A266489, A292395, A300734.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 15 2017