A357792 - OEIS

%I #26 Mar 14 2023 05:13:24

%S 1,1,1,3,7,20,60,189,613,2039,6918,23850,83315,294282,1049279,3771685,

%T 13653313,49730599,182130129,670274170,2477514172,9193599339,

%U 34237330355,127914531260,479318575375,1800971051420,6783809423496,25611913597250,96903193235645,367363376407250

%N a(n) = coefficient of x^n in A(x) = Sum_{n>=0} C(x)^n * (1 - C(x)^n)^n, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n)^n, which holds as a formal power series in x.

%C Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - C(x)^n)^n / (1 - C(x))^n, where C(x) = x + C(x)^2.

%H Paul D. Hanna, <a href="/A357792/b357792.txt">Table of n, a(n) for n = 0..400</a>

%F Given C(x) = x + C(x)^2, g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by:

%F (1) A(x) = Sum_{n>=0} C(x)^n * (1 - C(x)^n)^n.

%F (2) A(x) = Sum_{n>=1} (-1)^(n-1) * C(x)^(n*(n-1)) / (1 - C(x)^n)^n.

%F (3) A(x) = Sum_{n>=0} x^n * [ (1 - C(x)^n) / (1 - C(x)) ]^n.

%F (4) A(x) = Sum_{n>=1} -(-1/x)^n * C(x)^(n^2) / [ (1 - C(x)^n) / (1 - C(x)) ]^n.

%F a(n) ~ c * 2^(2*n) / n^(3/2), where c = 0.1930490961334149255878338532701052858837... - _Vaclav Kotesovec_, Mar 14 2023

%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 20*x^5 + 60*x^6 + 189*x^7 + 613*x^8 + 2039*x^9 + 6918*x^10 + 23850*x^11 + 83315*x^12 + ...

%e Let C = C(x) = x + C(x)^2, then

%e A(x) = 1 + C*(1 - C) + C^2*(1 - C^2)^2 + C^3*(1 - C^3)^3 + C^4*(1 - C(x)^4)^4 + C^5*(1 - C(x)^5)^5 + ... + C(x)^n * (1 - C(x)^n)^n + ...

%e also,

%e A(x) = 1 + x*(1) + x^2*(1 + C)^2 + x^3*(1 + C + C^2)^3 + x^4*(1 + C + C^2 + C^3)^4 + x^5*(1 + C + C^2 + C^3 + C^4)^5 + x^6*(1 + C + C^2 + C^3 + C^4 + C^5)^6 + ... + x^n*(1 + C + C^2 + C^3 + ... + C^(n-1))^n + ...

%e further,

%e A(x) = 1/(1 - C) - C^2/(1 - C^2)^2 + C^6/(1 - C^3)^3 - C^12/(1 - C^4)^4 + C^20/(1 - C^5)^5 + ... + (-1)^(n-1) * C(x)^(n*(n-1)) / (1 - C^n)^n + ...

%e where the related Catalan series, C(x) = (1 - sqrt(1 - 4*x))/2, begins:

%e C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 + ... + A000108(n)*x^(n+1) + ...

%e SPECIFIC VALUES.

%e The radius of convergence of the power series A(x) equals 1/4.

%e The power series A(x) converges at x = 1/4 to

%e A(1/4) = 1.578564238051657388445969550353857020762848420638921268996...

%e which equals the following sums:

%e (1) A(1/4) = Sum_{n>=0} (2^n - 1)^n / 2^(n*(n+1)),

%e (2) A(1/4) = Sum_{n>=1} (-1)^(n-1) * 2^n / (2^n - 1)^n.

%o (PARI) {a(n) = my(A=1, C = serreverse(x-x^2 + O(x^(n+2))));

%o A = sum(m=0,n, C^m * (1 - C^m)^m); polcoeff(A,n)}

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

%o (PARI) {a(n) = my(A=1, C = serreverse(x-x^2 + O(x^(n+2))));

%o A = sum(m=0,n, x^m * (1 - C^m)^m/(1 - C)^m); polcoeff(A,n)}

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

%o (PARI) {a(n) = my(A=1, C = serreverse(x-x^2 + O(x^(n+2))));

%o A = sum(m=1,n+1, (-1)^(m-1) * C^(m*(m-1)) / (1 - C^m)^m); polcoeff(A,n)}

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

%Y Cf. A357793, A000108.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Dec 14 2022