OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 5*A(x) = 1 + x + 4*A(x*A(x)).
(2) A(x) = 1 + Sum_{n>=0} 4^n * B^n(x) / 5^(n+1), where B(x) = x*A(x) and B^n(x) denotes the n-th iteration of B(x) with B^0(x) = x.
(3) 4*A(x) = 5*F(x) - x/F(x) - 1, where F(x) = A(x/F(x)) = x/Series_Reversion(x*A(x)).
a(n) ~ c * n^(n + 3/10 + 3*log(5/4)) / (exp(n) * log(5/4)^n), where c = 0.197680083007236432113... - Vaclav Kotesovec, Apr 06 2026
From Seiichi Manyama, Apr 06 2026: (Start)
Let b(n,k) = [x^n] A(x)^k.
b(0,1) = b(1,1) = 1; b(n,1) = 4 * Sum_{j=1..n-1} b(j,1) * b(n-j,j).
For k > 1, b(0,k) = 1; b(n,k) = (1/n) * Sum_{j=1..n} ((k+1)*j-n) * b(j,1) * b(n-j,k).
a(n) = b(n,1). (End)
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 48*x^3 + 912*x^4 + 22784*x^5 + 690496*x^6 + 24314624*x^7 + 969209152*x^8 + 42986764544*x^9 + ...
RELATED SERIES.
A(x*A(x)) = 1 + x + 5*x^2 + 60*x^3 + 1140*x^4 + 28480*x^5 + 863120*x^6 + 30393280*x^7 + 1211511440*x^8 + 53733455680*x^9 + ...
where A(x) = (1 + x)/5 + (4/5)*A(x*A(x)).
If F(x) = A(x/F(x)), then
F(x) = 1 + x + 3*x^2 + 38*x^3 + 723*x^4 + 18098*x^5 + 549086*x^6 + 19349740*x^7 + 771736003*x^8 + 34243429882*x^9 + ...
where 4*A(x) = 5*F(x) - x/F(x) - 1.
RELATED TABLE.
The table of coefficients in the n-th iteration of x*A(x) begins
n = 0: [1, 0, 0, 0, 0, 0, 0, ...];
n = 1: [1, 1, 4, 48, 912, 22784, 690496, ...];
n = 2: [1, 2, 10, 117, 2180, 53796, 1616880, ...];
n = 3: [1, 3, 18, 213, 3918, 95616, 2850492, ...];
n = 4: [1, 4, 28, 342, 6264, 151526, 4483036, ...];
n = 5: [1, 5, 40, 510, 9380, 225630, 6631512, ...];
n = 6: [1, 6, 54, 723, 13452, 322974, 9443868, ...];
n = 7: [1, 7, 70, 987, 18690, 449666, 13105372, ...];
...
in which the following sum along column k equals a(k)
a(1) = 1 = 1/5 + 1*4/5^2 + 1*4^2/5^3 + 1*4^3/5^4 + 1*4^4/5^5 + ...
a(2) = 4 = 0/5 + 1*4/5^2 + 2*4^2/5^3 + 3*4^3/5^4 + 4*4^4/5^5 + ...
a(3) = 48 = 0/5 + 4*4/5^2 + 10*4^2/5^3 + 18*4^3/5^4 + 28*4^4/5^5 + ...
a(4) = 912 = 0/5 + 48*4/5^2 + 117*4^2/5^3 + 213*4^3/5^4 + 342*4^4/5^5 + ...
a(5) = 22784 = 0/5 + 912*4/5^2 + 2180*4^2/5^3 + 3918*4^3/5^4 + 6264*4^4/5^5 + ...
a(6) = 690496 = 0/5 + 22784*4/5^2 + 53796*4^2/5^3 + 95616*4^3/5^4 + 151526*4^4/5^5 + ...
a(7) = 24314624 = 0/5 + 690496*4/5^2 + 1616880*4^2/5^3 + 2850492*4^3/5^4 + 4483036*4^4/5^5 + ...
...
PROG
(PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1+x + 4*subst(A, x, x*A +x*O(x^n)) - 4*A); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)}
upto(30)
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 05 2026
STATUS
approved