A352409 G.f. A(x) satisfies: [x^(n+1)] (1+x - x^2*A(x))^(n*(2*n+1)) = 0, for n >= 0.

1, 3, 45, 1267, 51597, 2761539, 182885885, 14415019395, 1316237331069, 136512958750979, 15842506286290173, 2033176597680449283, 285833727841312233725, 43677225803116362273795, 7207197437612731825348605, 1277141936892060488486787075

Offset: 0

Author

Paul D. Hanna, Mar 15 2022

Keywords

• nonn

Comments

Conjectures:

(1) a(3*n) = A039969(n) (mod 3) = A000108(n) (mod 3) for n >= 0, where A000108 is the Catalan sequence.

(2) a(3*n+1) = 0 (mod 3) for n >= 0.

(3) a(3*n+2) = 0 (mod 3) for n >= 0.

(4) a(n) = 1 (mod 2) for n >= 0.

Examples

			G.f.: A(x) = 1 + 3*x + 45*x^2 + 1267*x^3 + 51597*x^4 + 2761539*x^5 + 182885885*x^6 + 14415019395*x^7 + 1316237331069*x^8 + ...
Related table.
The table of coefficients of x^k in (1+x - x^2*A(x))^(n*(2*n+1)) begins:
n=0: [1,  0,    0,     0,       0,        0,         0, ...];
n=1: [1,  3,    0,   -14,    -153,    -4059,   -162214, ...];
n=2: [1, 10,   35,     0,    -825,   -17758,   -642015, ...];
n=3: [1, 21,  189,   847,       0,   -55818,  -1835218, ...];
n=4: [1, 36,  594,  5772,   32715,        0,  -4524660, ...];
n=5: [1, 55, 1430, 23100,  252450,  1762706,         0, ...];
n=6: [1, 78, 2925, 69836, 1179672, 14597856, 122423756, 0, ...];
...
in which a diagonal equals all zeros, illustrating that
[x^(n+1)] (1+x - x^2*A(x))^(n*(2*n+1)) = 0, for n >= 0.
Congruence modulo 3:
(1) The terms of this sequence appear to be divisible by 3 when the index is not divisible by 3:
a(3*n+k) = 0 (mod 3) for n >= 0 and k = 1 or 2.
(2) For the terms a(3*n), the residues modulo 3 begin:
a(3*n) (mod 3) = [1, 1, 2, 2, 2, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, ...],
which appears to be congruent to the Catalan sequence A000108 modulo 3; i.e.,
a(3*n) = binomial(2*n,n)/(n+1) (mod 3), for n >= 0.
The above conjectures have been verified for the initial 1201 terms of this sequence.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A352408, A039969, A000108.

Programs

PARI

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A;
A[#A] = polcoeff( (1+x - x^2*Ser(A))^(m*(2*m+1)) / (m*(2*m+1)) ,m+1););A[n+1]}
for(n=0,10,print1(a(n),", "))

Formula

Conjectures: g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) A(x) = 1 + x^3*A(x)^2 (mod 3),

(2) A(x) = C(x^3) (mod 3) = C(x)^3 (mod 3), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

a(n) ~ c * 2^(3*n) * (n-1)! / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2))))^n, where c = 0.87591174815917817179... - Vaclav Kotesovec, Mar 19 2022