A230134 Expansion of e.g.f. 1/(1 - sin(5*x))^(1/5).
1, 1, 6, 41, 456, 6301, 108576, 2207981, 52012416, 1390239481, 41593598976, 1376769180401, 49955931795456, 1971671764875541, 84095262825824256, 3854514200269774901, 188942180401957502976, 9863099585213327293681, 546266997049408050364416, 31993839349571172423492281
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 41*x^3/3! + 456*x^4/4! + 6301*x^5/5! +... O.g.f.: 1/(1-x - 5*1*2/2*x^2/(1-6*x - 5*2*7/2*x^2/(1-11*x - 5*3*12/2*x^2/(1-16*x - 5*4*17/2*x^2/(1-21*x - 5*5*22/2*x^2/(1-...)))))), a continued fraction.
Programs
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Mathematica
CoefficientList[Series[1/(1-Sin[5*x])^(1/5), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jan 03 2014 *) -
PARI
{a(n)=local(X=x+x*O(x^n)); n!*polcoeff((1-sin(5*X))^(-1/5), n)} for(n=0, 20, print1(a(n), ", ")) -
PARI
{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^(5/2)/subst(A^(5/2), x, -x)))); n!*polcoeff(A, n)} for(n=0, 20, print1(a(n), ", ")) -
PARI
a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j)); a008548(n) = prod(k=0, n-1, 5*k+1); a(n) = sum(k=0, n, a008548(k)*(5*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025
Formula
E.g.f. A(x) satisfies: A(x) = (cos(5*x/2) - sin(5*x/2))^(-2/5).
O.g.f.: 1/G(0) where G(k) = 1 - (5*k+1)*x - 5*(k+1)*(5*k+2)/2*x^2/G(k+1) [continued fraction formula from A144015 due to Sergei N. Gladkovskii].
a(n) ~ n! * sqrt(5+sqrt(5)) * Gamma(3/5) * 2^(n-9/10) * 5^n / (n^(3/5) * Pi^(n+7/5)). - Vaclav Kotesovec, Jan 03 2014
a(n) = Sum_{k=0..n} A008548(k) * (5*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025