OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
FORMULA
From Paul D. Hanna, Jul 27 2009: (Start)
G.f. satisfies: A(x) = A(x^3)*(1+x+x^2)/(1-x)^2.
Define TRISECTIONS: A(x) = T_0(x^3) + x*T_1(x^3) + x^2*T_2(x^3), then:
T_1(x)/T_0(x) = 3*(1 + 2*x)/(1 + 7*x + x^2) and
T_2(x)/T_0(x) = 3*(2 + x)/(1 + 7*x + x^2).
(End)
EXAMPLE
G.f.: A(x) = 1 + 3*x + 6*x^2 + 12*x^3 + 21*x^4 + 33*x^5 + 51*x^6 + ...
log(A(x)) = 3*x + 3*x^2/2 + 9*x^3/3 + 3*x^4/4 + 3*x^5/5 + 9*x^6/6 + ...
From Paul D. Hanna, Jul 27 2009: (Start)
TRISECTIONS begin:
T_0(x) = 1 + 12*x + 51*x^2 + 147*x^3 + 354*x^4 + 753*x^5 + ...
T_1(x) = 3 + 21*x + 75*x^2 + 201*x^3 + 462*x^4 + 948*x^5 + ...
T_2(x) = 6 + 33*x + 105*x^2 + 267*x^3 + 591*x^4 + 1176*x^5 + ...
(End)
MATHEMATICA
nmax = 50; CoefficientList[Series[Exp[Sum[3^(IntegerExponent[k, 3] + 1)*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 01 2024 *)
PROG
(PARI) {a(n)=local(L=sum(m=1, n, 3*3^valuation(m, 3)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}
(PARI) {a(n)=local(A=1+x); for(i=0, n\3, A=subst(A, x, x^3+x*O(x^n))*(1+x+x^2)/(1-x+x*O(x^n))^2); polcoeff(A, n)} \\ Paul D. Hanna, Jul 27 2009
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Jul 20 2009
STATUS
approved