A352702 - OEIS

%I #15 May 11 2024 11:07:10

%S 1,1,2,4,9,22,55,142,376,1011,2758,7614,21220,59630,168759,480533,

%T 1375676,3957075,11430582,33144264,96434321,281447954,823734157,

%U 2417092933,7109265120,20955593252,61893804180,183148075432,542885589115,1611809502764,4792612539375

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

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

%H Paul D. Hanna, <a href="/A352702/b352702.txt">Table of n, a(n) for n = 0..1200</a>

%H Ira M. Gessel, <a href="https://www.emis.de/journals/SLC/wpapers/s91vortrag/gessel3.pdf">The Amdeberhan-Konvalinka Conjecture and Symmetric Functions</a>, Séminaire Lotharingien Comb. (2024). See p. 83 of 109.

%F G.f. A(x) satisfies:

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

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

%e 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 + ...

%e where

%e (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 + ...

%e also

%e (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 + ...

%e which equals 1 - x*A(x).

%o (PARI) {a(n) = my(A=1+x); for(i=1,n,

%o A = (1 - (1 - 3*x - x^3*subst(A,x,x^3) + x*O(x^(n+1)))^(1/3))/x);

%o polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A107092, A352704, A352706.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Mar 29 2022