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A227544 Expansion of e.g.f. 1/(1 - sin(6*x))^(1/6).

%I A227544 #38 Jun 24 2025 10:01:53
%S A227544 1,1,7,55,721,11761,240247,5801095,162512161,5171130721,184337942887,
%T A227544 7275081518935,314918762166001,14834964193292881,755507853144691927,
%U A227544 41362173671901329575,2422478811455080626241,151132171549872325122241,10006051653759338150151367,700695219796759105368529015
%N A227544 Expansion of e.g.f. 1/(1 - sin(6*x))^(1/6).
%C A227544 Generally, for e.g.f. 1/(1-sin(p*x))^(1/p) we have a(n) ~ n! * 2^(n+3/p) * p^n / (Gamma(2/p) * n^(1-2/p) * Pi^(n+2/p)). - _Vaclav Kotesovec_, Jan 03 2014
%F A227544 E.g.f. A(x) satisfies:
%F A227544 (1) A(x) = (cos(3*x) - sin(3*x))^(-1/3).
%F A227544 (2) A(x)^3/A(-x)^3 = 1/cos(6*x) + tan(6*x).
%F A227544 (3) A(x) = exp( Integral A(x)^3/A(-x)^3 dx ).
%F A227544 O.g.f.: 1/G(0) where G(k) = 1 - (6*k+1)*x - 6*(k+1)*(3*k+1)*x^2/G(k+1) [continued fraction formula from A144015 due to _Sergei N. Gladkovskii_].
%F A227544 a(n) ~ n! * 2^(2*n+1/2) * 3^n / (Gamma(1/3) * n^(2/3) * Pi^(n+1/3)). - _Vaclav Kotesovec_, Jan 03 2014
%F A227544 a(n) = Sum_{k=0..n} A008542(k) * (6*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - _Seiichi Manyama_, Jun 24 2025
%e A227544 E.g.f.: A(x) = 1 + x + 7*x^2/2! + 55*x^3/3! + 721*x^4/4! + 11761*x^5/5! + ...
%e A227544 where A(x)^3 = 1 + 3*x + 27*x^2/2! + 297*x^3/3! + 4617*x^4/4! + 87723*x^5/5! + ...
%e A227544 and 1/A(x)^3 = 1 - 3*x - 9*x^2/2! + 27*x^3/3! + 81*x^4/4! - 243*x^5/5! + ...
%e A227544 which illustrates 1/A(x)^3 = cos(3*x) - sin(3*x).
%e A227544 O.g.f.: 1/(1-x - 6*1*1*x^2/(1-7*x - 6*2*4*x^2/(1-13*x - 6*3*7*x^2/(1-19*x - 6*4*10*x^2/(1-25*x - 6*5*13*x^2/(1-...)))))), a continued fraction.
%t A227544 CoefficientList[Series[1/(1-Sin[6*x])^(1/6), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jan 03 2014 *)
%o A227544 (PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(3*X)-sin(3*X))^(-1/3), n)}
%o A227544 for(n=0,20,print1(a(n),", "))
%o A227544 (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(intformal(A^3/subst(A^3, x, -x)))); n!*polcoeff(A, n)}
%o A227544 for(n=0,20,print1(a(n),", "))
%o A227544 (PARI) a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
%o A227544 a008542(n) = prod(k=0, n-1, 6*k+1);
%o A227544 a(n) = sum(k=0, n, a008542(k)*(6*I)^(n-k)*a136630(n, k)); \\ _Seiichi Manyama_, Jun 24 2025
%Y A227544 Cf. A001586 (p=2), A007788 (p=3), A144015 (p=4), A230134 (p=5), A235128 (p=7), A230114 (p=8).
%Y A227544 Cf. A008542, A136630.
%K A227544 nonn
%O A227544 0,3
%A A227544 _Paul D. Hanna_, Dec 20 2013