A352702 G.f. A(x) satisfies: (1 - x*A(x))^3 = 1 - 3*x - x^3*A(x^3).

1, 1, 2, 4, 9, 22, 55, 142, 376, 1011, 2758, 7614, 21220, 59630, 168759, 480533, 1375676, 3957075, 11430582, 33144264, 96434321, 281447954, 823734157, 2417092933, 7109265120, 20955593252, 61893804180, 183148075432, 542885589115, 1611809502764, 4792612539375

Offset: 0

Author

Paul D. Hanna, Mar 29 2022

Keywords

• nonn

Comments

Essentially an unsigned version of A107092 (after dropping the initial term).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 55*x^6 + 142*x^7 + 376*x^8 + 1011*x^9 + 2758*x^10 + 7614*x^11 + ...
where
(1 - x*A(x))^3 = 1 - 3*x - x^3 - x^6 - 2*x^9 - 4*x^12 - 9*x^15 - 22*x^18 - 55*x^21 - 142*x^24 - 376*x^27 - 1011*x^30 + ...
also
(1 - 3*x - x^3*A(x^3))^(1/3) = 1 - x - x^2 - 2*x^3 - 4*x^4 - 9*x^5 - 22*x^6 - 55*x^7 - 142*x^8 - 376*x^9 - 1011*x^10 + ...
which equals 1 - x*A(x).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1200
• Ira M. Gessel, The Amdeberhan-Konvalinka Conjecture and Symmetric Functions, Séminaire Lotharingien Comb. (2024). See p. 83 of 109.

Crossrefs

Cf. A107092, A352704, A352706.

Programs

PARI

{a(n) = my(A=1+x); for(i=1,n,
A = (1 - (1 - 3*x - x^3*subst(A,x,x^3) + x*O(x^(n+1)))^(1/3))/x);
polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) (1 + x*A(-x))^3 = 1 + 3*x + x^3*A(-x^3).

(2) A(x) = (1 - (1 - 3*x - x^3*A(x^3))^(1/3))/x.