A372529 Expansion of g.f. A(x) satisfying A(x) = A(x^3 + x^4) / A(x^2).

1, 1, -1, 0, 4, 2, -6, -5, 3, 2, -7, -9, 6, 29, 42, 25, 12, 49, 71, -26, -198, -248, -176, -226, -456, -456, 68, 622, 692, 694, 1224, 1637, 817, -1064, -2996, -4686, -6860, -9552, -10808, -7816, -33, 10534, 24816, 48032, 80896, 113786, 140418, 168927, 200612, 209210

Offset

1,5

Comments

Compare to C(x) = C( x^3 + 3*x*C(x)^3 ) / C( x^2 + 2*x*C(x)^2 ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Conjecture: lim_{n_>oo} abs(a(n))^(-1/n) = r where r = 0.75487766624669276... is the real root of x^3 + x^2 - 1 = 0.

Positions where terms change from nonnegative to negative (or vice versa) are: [1, 3, 4, 7, 9, 11, 13, 20, 27, 34, 42, 66, 90, 113, 138, 222, 306, 383, 470, 756, 1044, 1309, 1610, 2590, ...].

Links

Paul D. Hanna, Table of n, a(n) for n = 1..3351

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = A(x^3 + x^4) / A(x^2).

(2) A(x) = A(x^3 + x^4)/A(x^6 + x^8) * A(x^4).

(3) A(x) = A(x^3 + x^4)/A(x^6 + x^8) * A(x^12 + x^16)/A(x^24 + x^32) * A(x^16).

Example

G.f. A(x) = x + x^2 - x^3 + 4*x^5 + 2*x^6 - 6*x^7 - 5*x^8 + 3*x^9 + 2*x^10 - 7*x^11 - 9*x^12 + 6*x^13 + 29*x^14 + 42*x^15 + 25*x^16 + ...
where A(x) = A(x^3 + x^4) / A(x^2).
RELATED SERIES.
A(x^3 + x^4) = A(x)*A(x^2) = x^3 + x^4 + x^6 + 2*x^7 + x^8 - x^9 - 3*x^10 - 3*x^11 - x^12 + 4*x^15 + 20*x^16 + 40*x^17 + 42*x^18 + ...
Let R(x) be the series reversion of A(x), A(R(x)) = x, then
R(x) = x - x^2 + 3*x^3 - 10*x^4 + 34*x^5 - 128*x^6 + 500*x^7 - 2008*x^8 + 8282*x^9 - 34795*x^10 + 148438*x^11 - 641463*x^12 + ...
where A( R(x)^3 + R(x)^4 ) = x * A( R(x)^2 ).
SPECIFIC VALUES.
A(t) = 1 at t = (sqrt(5) - 1)/2 = 0.618033988749...
A(2/3) = 1.15675491342220964379947511396627403569871267716969804587461...
where A(2/3) = A(40/81) / A(4/9)
also, A(2/3) = A(40/81) / A(832/6561) * A(16/81).
A(1/2) = 0.72139287697996766146675581382502493921019742715562896572242...
where A(1/2) = A(3/16) / A(1/4)
also, A(1/2) = A(3/16) / A(5/256) * A(1/16).
A(1/3) = 0.42324964669257285241360643415569236002723373572120164572426...
where A(1/3) = A(4/81) / A(1/9)
also, A(1/3) = A(4/81) / A(10/6561) * A(1/81).
A(1/4) = 0.30083841639573427800841546749116173913453320737719157300262...
where A(1/4) = A(5/256) / A(1/16)
also, A(1/4) = A(5/256) / A(17/65536) * A(1/256).
A(1/5) = 0.23331997181413989512607131768669995870957327606980911323852...
where A(1/5) = A(6/625) / A(1/25).

Prog

(PARI) {a(n) = my(Ax = x); for(m=1, n, Ax = truncate(Ax);
Ax = subst(Ax, x, x^3 + x^4 +x^3*O(x^m)) / subst(Ax, x, x^2 +x^3*O(x^m)) ); polcoeff(Ax, n)}
for(n=1, 50, print1(a(n), ", "))

Crossrefs

Sequence in context: A176836 A329287 A368665 * A021705 A228084 A344123

Adjacent sequences: A372526 A372527 A372528 * A372530 A372531 A372532

Keyword

sign

Author

Paul D. Hanna, May 25 2024

Status

approved