A292394 G.f. A(x) satisfies: [x^n] A( x/A(x)^(n^2) ) = 0 for n>1.

1, 1, 4, 63, 2160, 138025, 15073596, 2572772048, 642195593408, 224210211292629, 105538947969000700, 64968095569659693840, 51052911516803277805536, 50200764753583055762144696, 60726306448248686394356127312, 89059256525528149546284788980800, 156362202771639017446505434119944960, 325036812372664804706525309628909742683

Offset

0,3

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Example

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 +...
The table of coefficients in A(x/A(x)^(n^2)) begins:
n=0: [1, 1, 4, 63, 2160, 138025, 15073596, 2572772048, 642195593408, ...];
n=1: [1, 1, 3, 52, 1895, 126554, 14211827, 2465030616, 621409924326, ...];
n=2: [1, 1, 0, 25, 1228, 96586, 11876160, 2164945551, 562519626368, ...];
n=3: [1, 1, -5, 0, 483, 59116, 8696349, 1732963772, 474868166564, ...];
n=4: [1, 1, -12, 7, 0, 25509, 5385244, 1244512060, 370970521936, ...];
n=5: [1, 1, -21, 88, -165, 0, 2430241, 766004308, 263319541888, ...];
n=6: [1, 1, -32, 297, -660, -24926, 0, 342132167, 161945375456, ...];
n=7: [1, 1, -45, 700, -3377, -64394, -1721889, 0, 73311075962, ...];
n=8: [1, 1, -60, 1375, -12112, -120679, -1685572, -241494912, 0, ...];
n=9: [1, 1, -77, 2412, -33345, -137186, 3227751, -394368976, -60181947214, 0, ...]; ...
such that the main diagonal consists of all zeros after the initial terms.
The values for a(n)/n, for n>=1, begin:
[1, 2, 21, 540, 27605, 2512266, 367538864, 80274449176, 24912245699181, 10553894796900070, 5906190506332699440, ...].
The values for a(n)/n^2, for n>=1, form A292395 and begin:
[1, 1, 7, 135, 5521, 418711, 52505552, 10034306147, 2768027299909, 1055389479690007, 536926409666609040, ...].

Prog

(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]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A266489, A292395, A300734.

Sequence in context: A274392 A237644 A024256 * A351779 A203512 A354653

Adjacent sequences: A292391 A292392 A292393 * A292395 A292396 A292397

Keyword

nonn

Author

Paul D. Hanna, Sep 15 2017

Status

approved