OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..378
FORMULA
G.f. A(x) satisfies:
(1) [x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2) * A(x)^(3*n+2) for n >= 1.
(2) A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)).
(3) A'(x) = A(x) * (1 + x/(1 - A(x))) / (3*x).
(4) A(x) = exp( Integral (1 + x/(1 - A(x))) / (3*x) dx ).
a(n) ~ c * 3^n * n! * n^(2/3), where c = 0.09232038797888963484135336... - Vaclav Kotesovec, Nov 16 2023
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 309*x^4 + 5262*x^5 + 108894*x^6 + 2618718*x^7 + 71246145*x^8 + ...
such that A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)).
Related table.
The table of coefficients of x^k in A(x)^(3*n+2) begins:
n=0: [1, 2, 7, 54, 675, 11286, 230742, ...];
n=1: [1, 5, 25, 190, 2210, 34981, 688635, ...];
n=2: [1, 8, 52, 416, 4642, 69872, 1322848, ...];
n=3: [1, 11, 88, 759, 8349, 120549, 2195886, ...];
n=4: [1, 14, 133, 1246, 13790, 193060, 3391017, ...];
n=5: [1, 17, 187, 1904, 21505, 295154, 5017618, ...];
n=6: [1, 20, 250, 2760, 32115, 436524, 7217250, ...]; ...
in which the following pattern holds:
[x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2) * A(x)^(3*n+2), n >= 1,
as illustrated by
[x^1] A(x)^2 = 2 = [x^0] 2*A(x)^2 = 2*1;
[x^2] A(x)^5 = 25 = [x^1] 5*A(x)^5 = 5*5;
[x^3] A(x)^8 = 416 = [x^2] 8*A(x)^8 = 8*52;
[x^4] A(x)^11 = 8349 = [x^3] 11*A(x)^11 = 11*759;
[x^5] A(x)^14 = 193060 = [x^4] 14*A(x)^14 = 14*13790;
[x^6] A(x)^17 = 5017618 = [x^5] 17*A(x)^17 = 17*295154; ...
PROG
(PARI) /* Using A(x) = 1 + x*A(x)/(A(x) - 3*x*A'(x)) */
{a(n) = my(A=1); for(i=1, n, A = 1 + x*A/(A - 3*x*A' + x*O(x^n)) );
polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Using [x^n] A(x)^(3*n+2) = [x^(n-1)] (3*n+2)*A(x)^(3*n+2) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff((x*Ser(A)^(3*(#A-2)+2) - Ser(A)^(3*(#A-2)+2)/(3*(#A-2)+2)), #A-1)); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 08 2022
STATUS
approved