OFFSET
1,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..2201
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A(x^3) + 3*A(x^4).
(2) A(x)^9 = A(x^9) + 9*A(x^3)^2*A(x^4) + 27*A(x^3)*A(x^4)^2 + 30*A(x^12) + 81*A(x^16).
EXAMPLE
G.f.: A(x) = x + x^2 - x^3 + 2*x^4 - 4*x^5 + 10*x^6 - 24*x^7 + 60*x^8 - 156*x^9 + 415*x^10 - 1126*x^11 + 3099*x^12 - 8630*x^13 + 24266*x^14 - 68794*x^15 + ...
A(x)^3 = x^3 + 3*x^4 + x^6 + 3*x^8 - x^9 - x^12 - 4*x^15 + 6*x^16 + 10*x^18 - 12*x^20 - 24*x^21 + 90*x^24 - 156*x^27 - 72*x^28 + 415*x^30 + 180*x^32 - 1126*x^33 + 2631*x^36 + ...
where A(x)^3 = A(x^3) + 3*A(x^4).
SPECIFIC VALUES.
A(t) = 2/5 at t = 0.317791549876656168371238513970934428132062497551072...
A(t) = 1/3 at t = 0.272012573923351948160518716810476732450297934080149...
A(t) = 1/4 at t = 0.211741163482363760573873114118831096471558139140673...
A(t) = 1/5 at t = 0.173689492982350976627014476261587172865373530722099...
A(1/4) = 0.302314652721872148883870605812559015254486733843572...
where A(1/4)^3 = A(1/64) + 3*A(1/256).
A(1/5) = 0.234354127007807369375669036669209786321308301358256...
where A(1/5)^3 = A(1/125) + 3*A(1/625).
A(1/6) = 0.190997197977808986635998271807026082408099105202937...
A(1/8) = 0.139067491696071081958857480435275933590079492123770...
A(1/27) = 0.038361482183250487316728449871172355741923939485303...
A(1/64) = 0.015865441552000659797364352852103507166739710300423...
A(1/81) = 0.012496258474545183941754142582223891156858749129310...
A(1/125) = 0.0080634960635000944831333442317108094366638538262799...
A(1/256) = 0.0039214496464762303998150751827603932832138166123905...
A(1/625) = 0.0016025559170654240904812111717421435967661263615758...
PROG
(PARI) {a(n) = my(A, V=[0, 1]); for(i=1, n+1, V=concat(V, 0); A=Ser(V);
V[#V] = polcoef( (subst(A, x, x^3) + 3*subst(A, x, x^4) - A^3 )/3, #V+1) ); polcoef(A, n)}
for(n=1, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Nov 28 2024
STATUS
approved