A384821 G.f. A(x) satisfies -1/x = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+2).
1, 2, 5, 22, 91, 416, 1978, 9738, 49181, 253572, 1328528, 7053672, 37866294, 205188765, 1120824743, 6165155890, 34119043994, 189839648588, 1061344406923, 5959197795092, 33588952625106, 189986944364176, 1078034452020854, 6134848540680166, 35005230073846833, 200229444332667654
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 5*x^2 + 22*x^3 + 91*x^4 + 416*x^5 + 1978*x^6 + 9738*x^7 + 49181*x^8 + 253572*x^9 + 1328528*x^10 + ... SPECIFIC VALUES. A(t) = 2 at t = 0.162924020448782314256916956456618618555937137963260... A(t) = 9/5 at t = 0.15713093477961462528780113190237390843002535981643... A(t) = 8/5 at t = 0.14467881602482935797425598908263109752382579929421... A(t) = 3/2 at t = 0.13461615563760120581581313629107981605312435881819... A(t) = 4/3 at t = 0.10915621052082212882653574706851509193398803739915... A(1/7) = 1.5793911503434252677981671019480264164820055324466... A(1/8) = 1.4268350851974567615394958810072981944850896947894... A(1/9) = 1.3435470274993477728207146854713823085043981519155... A(1/10) = 1.2892440747830023480637465318368592024118039394009... A(1/11) = 1.2505209808081799972669805855553805055082827658365...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..401
Programs
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PARI
{a(n) = my(A=[1,2,0]); for(i=1, n, A = concat(A, 0); A[#A-1] = polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+2) ), #A-4)); A[n+1]} for(n=0, 30, print1(a(n), ", "))
Formula
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) -1/x = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+2).
(2) -x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-1)*(n-2)) / (1 - x^n)^(n-2).
a(n) ~ c * d^n / n^(3/2), where d = 6.07021478936467894926862346663483720359... and c = 0.6881950589132830412100382237325446... - Vaclav Kotesovec, Jun 11 2025