A229044 G.f. A(x) satisfies: [x^(n+1)] A(x)^(n^2) = 0 for n>=0.

1, 1, 0, -1, -6, -78, -1544, -40605, -1328178, -51857806, -2350025232, -121120896906, -6991877399100, -446673990116508, -31277285155060464, -2381645560450404989, -195914136385421694954, -17312472044077536945630, -1635541992950202705979424, -164494265246550280147797438

Offset

0,5

Links

Table of n, a(n) for n=0..19.

Formula

a(n) is odd iff n+1 is a power of 2 (conjecture).

G.f. A(x) satisfies the following relationes.

(1) [x^(n+1)] A(x)^(n^2) = 0 for n>=0.

(2) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A185072.

(3) A(x)/A(x)' is the g.f. of A305144. - Paul D. Hanna, Oct 23 2020

Example

G.f.: A(x) = 1 + x - x^3 - 6*x^4 - 78*x^5 - 1544*x^6 - 40605*x^7 -...
Coefficients of x^k in the square powers A(x)^(n^2) of g.f. A(x) begin:
n=1: [1, 1,   0,   -1,    -6,    -78,   -1544,   -40605,  -1328178, ...];
n=2: [1, 4,   6,    0,   -35,   -396,   -7182,  -181824,  -5817510, ...];
n=3: [1, 9,  36,   75,     0,  -1260,  -21408,  -499203, -15299145, ...];
n=4: [1,16, 120,  544,  1484,      0,  -52656, -1202240, -34269906, ...];
n=5: [1,25, 300, 2275, 11900,  40680,       0, -2557775, -73526475, ...];
n=6: [1,36, 630, 7104, 57429, 345204, 1430418,        0,-142432290, ...];
n=7: [1,49,1176,18375,209230,1833678,12546744, 61418175,         0, ...];
n=8: [1,64,2016,41600,630960,7470336,71271616,549420288,3113335320, 0, ...]; ...
where the coefficients of x^(n+1) in A(x)^(n^2) all equal zero for n>=0.
Related expansions.
A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A185072:
G(x) = 1 + x - 2*x^2 + 6*x^3 - 28*x^4 + 70*x^5 - 1446*x^6 -...
A(x)'/A(x) = 1 - x - 2*x^2 - 21*x^3 - 364*x^4 - 8830*x^5 - 273972*x^6 - 10313037*x^7 - 455135384*x^8 - 22995056286*x^9 - 1307053358940*x^10 - ...
A(x)/A(x)' = 1 + x + 3*x^2 + 26*x^3 + 417*x^4 + 9726*x^5 + 295000*x^6 + 10946172*x^7 + 478392123*x^8 + ... + A305144(n)*x^n + ...

Prog

(PARI) {a(n)=local(A=[1, 1]); for(k=1, n, A=concat(A, 0); A[#A]=-polcoeff((Ser(A) +O(x^(k+2)))^(k^2)/(k^2), k+1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A185072, A305144, A230218, A229041, A171791.

Sequence in context: A132866 A279444 A094419 * A300874 A049209 A162656

Adjacent sequences: A229041 A229042 A229043 * A229045 A229046 A229047

Keyword

sign

Author

Paul D. Hanna, Sep 12 2013

Status

approved