OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..400
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1 + 5*A(x) = Sum_{n>=0} (x + 4*A(x)^n)^n.
(2) 1 + 5*A(x) = Sum_{n>=0} 4^n * A(x)^(n^2) / (1 - x*A(x)^n)^(n+1).
a(n) ~ c * d^n / n^(3/2), where d = 8.4161217021922017915975... and c = 0.05404561143143091714667... - Vaclav Kotesovec, Jan 22 2025
EXAMPLE
G.f.: A(x) = x + x^2 + 9*x^3 + 33*x^4 + 229*x^5 + 1381*x^6 + 9169*x^7 + 63225*x^8 + 440257*x^9 + 3164657*x^10 + 22959933*x^11 + 169226413*x^12 + ...
where
1 + 5*A(x) = 1 + (x + 4*A(x)) + (x + 4*A(x)^2)^2 + (x + 4*A(x)^3)^3 + (x + 4*A(x)^4)^4 + (x + 4*A(x)^5)^5 + ...
also
1 + 5*A(x) = 1/(1 - x) + 4*A(x)/(1 - x*A(x))^2 + 4^2*A(x)^4/(1 - x*A(x)^2)^3 + 4^3*A(x)^9/(1 - x*A(x)^3)^4 + 4^4*A(x)^16/(1 - x*A(x)^4)^5 + ...
SPECIFIC VALUES.
A(t) = 1/6 at t = 0.113638591205494141262966261697414535895471468773250...
where 11/6 = Sum_{n>=0} (t + 4/6^n)^n.
A(t) = 1/7 at t = 0.105910494802205532838470290251326622991314014012676...
A(t) = 1/8 at t = 0.097962467685246150583774699711035184856986586455937...
where 13/8 = Sum_{n>=0} (t + 4/8^n)^n.
A(t) = 1/10 at t = 0.083911453795561366662465573676038011336636918103355...
where 3/2 = Sum_{n>=0} (t + 4/10^n)^n.
A(1/9) = 0.15766254452616863716010326059518744322051352273284...
where 1 + 5*A(1/9) = Sum_{n>=0} (1/9 + 4*A(1/9)^n)^n.
A(1//10) = 0.12921529144780914878336865509665225412719293526704...
A(1/12) = 0.099081728855742334696126021511606257573363030424504...
PROG
(PARI) {a(n) = my(V=[0, 1], A=x); for(i=1, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(n=1, #V, (x + 4*A^n)^n ) - 5*A, #V-1) ); V[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 11 2025
STATUS
approved