A378258 - OEIS

%I #6 Dec 03 2024 12:24:52

%S 1,3,22,219,2530,31830,422652,5820099,82226522,1183598754,17277560996,

%T 254957527134,3795103401236,56900877433356,858486641574392,

%U 13025190180880851,198640087778490698,3043793354165966442,46845889623804326292,723896233987491887466,11227150555314722435580

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

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

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

%C First negative term is at a(58).

%H Paul D. Hanna, <a href="/A378258/b378258.txt">Table of n, a(n) for n = 1..300</a>

%F 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).

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

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

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

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

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

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

%e 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 + ...

%e where A(x) = B(x) + 5*A(x)^2 with B(A(x) - A(x)^2) = x.

%e RELATED SERIES.

%e The Catalan series C(x) = x + C(x)^2 begins

%e 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 + ...

%e 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 + ...

%e 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 + ...

%e 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 + ...

%e Let B(x) satisfy B(A(x) - A(x)^2) = x, then

%e 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 + ...

%e 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 + ...

%o (PARI) \\ [x^n] A(A(x) - A(x)^2) = 5^(n-1)*A000108(n-1)

%o {A000108(n) = binomial(2*n+1,n)/(2*n+1)}

%o {a(n) = my(V=[0,1],A,m); for(i=1,n, V = concat(V,0); A = Ser(V); m = #V-1;

%o V[m+1] = ( 5^(m-1)*A000108(m-1) - polcoef(subst(A,x,A-A^2),m) )/2 ); V[n+1]}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A000108.

%K sign,new

%O 1,2

%A _Paul D. Hanna_, Dec 03 2024