A300734 G.f. A(x) satisfies: [x^n] A( x/A(x)^(2*n^2) ) = 0 for n>1.

1, 1, 8, 261, 18400, 2383800, 523322208, 179218487938, 89683747547520, 62708998782383118, 59089857758999295000, 72802718522742159383487, 114483047893936622981751744, 225239782463528942954480067562, 545115039824232864946208073815480, 1599336782706815943896682999835671375, 5617211714811070144606976491531059082752

Offset

0,3

Comments

Conjecture: n^2 divides a(n) for n>=1.

Links

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

Example

The table of coefficients in A(x/A(x)^(2*n^2)) begins:
n=0: [1, 1, 8, 261, 18400, 2383800, 523322208, 179218487938, ...];
n=1: [1, 1, 6, 216, 16180, 2187188, 493439464, 171718406096, ...];
n=2: [1, 1, 0, 105, 10568, 1672006, 412368512, 150825821184, ...];
n=3: [1, 1, -10, 0, 4228, 1024056, 301838400, 120746446648, ...];
n=4: [1, 1, -24, 21, 0, 439248, 186683808, 86739931322, ...];
n=5: [1, 1, -42, 336, -1500, 0, 84158448, 53438819528, ...];
n=6: [1, 1, -64, 1161, -5240, -403722, 0, 23924633200, ...];
n=7: [1, 1, -90, 2760, -26092, -1014644, -61791496, 0, ...];
n=8: [1, 1, -120, 5445, -94112, -1888744, -72316832, -16993280158, 0, ...]; ...
such that the main diagonal consists of all zeros after the initial terms.
The values a(n)/n^2 for n>=1 begin:
[1, 2, 29, 1150, 95352, 14536728, 3657520162, 1401308555430, 774185170152878, ...].

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)^(2*(#A-1)^2)))[#A]); A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A292394.

Sequence in context: A230727 A272392 A162083 * A081058 A003386 A299328

Adjacent sequences: A300731 A300732 A300733 * A300735 A300736 A300737

Keyword

nonn

Author

Paul D. Hanna, Mar 11 2018

Status

approved