A378258 G.f. satisfies A(x) = B(x) + 5*A(x)^2 where B(A(x) - A(x)^2) = x.

1, 3, 22, 219, 2530, 31830, 422652, 5820099, 82226522, 1183598754, 17277560996, 254957527134, 3795103401236, 56900877433356, 858486641574392, 13025190180880851, 198640087778490698, 3043793354165966442, 46845889623804326292, 723896233987491887466, 11227150555314722435580

Offset

1,2

Comments

Conjecture: a(n) is odd iff n is a power of 2.

Conjecture: a(n) is not congruent to 1 nor 5 (mod 6) for n > 1.

First negative term is at a(58).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas, wherein C(x) is the g.f. of the Catalan numbers (A000108 with offset 1).

(1) A(x) = B(x) + 5*A(x)^2 where B(A(x) - A(x)^2) = x.

(2) A(x) = C(5*B(x))/5 where B(A(x) - A(x)^2) = x and C(x) = x + C(x)^2.

(3) A(A(x) - A(x)^2) = C(5*x)/5 where C(x) = x + C(x)^2.

(4) A(x) = C(A(x) - A(x)^2) where C(x) = x + C(x)^2 (trivial).

(5) A(B(x)) = C(x) = x + C(x)^2 where B(A(x) - A(x)^2) = x.

(6) B(B(x)) = 5*x - 4*C(x) where B(A(x) - A(x)^2) = x and C(x) = x + C(x)^2.

Example

G.f.: A(x) = x + 3*x^2 + 22*x^3 + 219*x^4 + 2530*x^5 + 31830*x^6 + 422652*x^7 + 5820099*x^8 + 82226522*x^9 + 1183598754*x^10 + ...
where A(x) = B(x) + 5*A(x)^2 with B(A(x) - A(x)^2) = x.
RELATED SERIES.
The Catalan series C(x) = x + C(x)^2 begins
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + ... + A000108(n-1)*x^n + ...
A(x)^2 = x^2 + 6*x^3 + 53*x^4 + 570*x^5 + 6858*x^6 + 88476*x^7 + 1195565*x^8 + 16684770*x^9 + 238312766*x^10 + 3462822372*x^11 + ...
A(x) - A(x)^2 = x + 2*x^2 + 16*x^3 + 166*x^4 + 1960*x^5 + 24972*x^6 + 334176*x^7 + 4624534*x^8 + 65541752*x^9 + 945285988*x^10 + ...
where A(A(x) - A(x)^2) = C(5*x)/5 = x + 5*x^2 + 50*x^3 + 625*x^4 + 8750*x^5 + 131250*x^6 + 2062500*x^7 + ... + 5^(n-1)*A000108(n-1)*x^n + ...
Let B(x) satisfy B(A(x) - A(x)^2) = x, then
B(x) = x - 2*x^2 - 8*x^3 - 46*x^4 - 320*x^5 - 2460*x^6 - 19728*x^7 - 157726*x^8 - 1197328*x^9 - 7965076*x^10 - 36550864*x^11 + 79262804*x^12 + ...
where B(B(x)) = 5*x - 4*C(x) = x - 4*x^2 - 8*x^3 - 20*x^4 - 56*x^5 - 168*x^6 - 528*x^7 - 1716*x^8 + ... + -4*A000108(n-1)*x^n + ...

Prog

(PARI) \\ [x^n] A(A(x) - A(x)^2) = 5^(n-1)*A000108(n-1)
{A000108(n) = binomial(2*n+1, n)/(2*n+1)}
{a(n) = my(V=[0, 1], A, m); for(i=1, n, V = concat(V, 0); A = Ser(V); m = #V-1;
V[m+1] = ( 5^(m-1)*A000108(m-1) - polcoef(subst(A, x, A-A^2), m) )/2 ); V[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A000108.

Sequence in context: A340474 A079489 A190526 * A141152 A372250 A066573

Adjacent sequences: A378255 A378256 A378257 * A378259 A378260 A378261

Keyword

sign,new

Author

Paul D. Hanna, Dec 03 2024

Status

approved