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 A357792 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). 2
 1, 1, 1, 3, 7, 20, 60, 189, 613, 2039, 6918, 23850, 83315, 294282, 1049279, 3771685, 13653313, 49730599, 182130129, 670274170, 2477514172, 9193599339, 34237330355, 127914531260, 479318575375, 1800971051420, 6783809423496, 25611913597250, 96903193235645, 367363376407250 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Related identity: 0 = Sum_{n=-oo..+oo} x^n * (1 - x^n)^n, which holds as a formal power series in x. 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. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..400 FORMULA Given C(x) = x + C(x)^2, g.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by: (1) A(x) = Sum_{n>=0} C(x)^n * (1 - C(x)^n)^n. (2) A(x) = Sum_{n>=1} (-1)^(n-1) * C(x)^(n*(n-1)) / (1 - C(x)^n)^n. (3) A(x) = Sum_{n>=0} x^n * [ (1 - C(x)^n) / (1 - C(x)) ]^n. (4) A(x) = Sum_{n>=1} -(-1/x)^n * C(x)^(n^2) / [ (1 - C(x)^n) / (1 - C(x)) ]^n. a(n) ~ c * 2^(2*n) / n^(3/2), where c = 0.1930490961334149255878338532701052858837... - Vaclav Kotesovec, Mar 14 2023 EXAMPLE 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 + ... Let C = C(x) = x + C(x)^2, then 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 + ... also, 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 + ... further, 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 + ... where the related Catalan series, C(x) = (1 - sqrt(1 - 4*x))/2, begins: 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) + ... SPECIFIC VALUES. The radius of convergence of the power series A(x) equals 1/4. The power series A(x) converges at x = 1/4 to A(1/4) = 1.578564238051657388445969550353857020762848420638921268996... which equals the following sums: (1) A(1/4) = Sum_{n>=0} (2^n - 1)^n / 2^(n*(n+1)), (2) A(1/4) = Sum_{n>=1} (-1)^(n-1) * 2^n / (2^n - 1)^n. PROG (PARI) {a(n) = my(A=1, C = serreverse(x-x^2 + O(x^(n+2)))); A = sum(m=0, n, C^m * (1 - C^m)^m); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n) = my(A=1, C = serreverse(x-x^2 + O(x^(n+2)))); A = sum(m=0, n, x^m * (1 - C^m)^m/(1 - C)^m); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n) = my(A=1, C = serreverse(x-x^2 + O(x^(n+2)))); A = sum(m=1, n+1, (-1)^(m-1) * C^(m*(m-1)) / (1 - C^m)^m); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A357793, A000108. Sequence in context: A132364 A110490 A132868 * A361625 A056783 A320735 Adjacent sequences: A357789 A357790 A357791 * A357793 A357794 A357795 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 14 2022 STATUS approved

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Last modified September 16 22:04 EDT 2024. Contains 375979 sequences. (Running on oeis4.)