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A144015
Expansion of e.g.f. 1/(1 - sin(4*x))^(1/4).
7
1, 1, 5, 29, 265, 3001, 42125, 696149, 13296145, 287706481, 6959431445, 186061833869, 5448382252825, 173418192216361, 5961442393047965, 220112963745653189, 8687730877758518305, 365023930617143804641, 16266420334783460443685, 766297734521812843642109
OFFSET
0,3
COMMENTS
Row sums of A186492 - Peter Bala, Feb 22 2011.
FORMULA
E.g.f. A(x) satisfies:
(1) A(x) = (cos(2*x) - sin(2*x))^(-1/2).
(2) A(x)^2/A(-x)^2 = 1/cos(4*x) + tan(4*x).
(3) A(x) = exp( Integral A(x)^2/A(-x)^2 dx).
(4) A'(x) = A(x)^3/A(-x)^2 with A(0) = 1.
(5) A(x) = 1/sqrt(1 - 2*Series_Reversion( Integral 1/sqrt(1+4*x-4*x^2) dx )).
G.f.: 1/G(0) where G(k) = 1 - x*(4*k+1) - 4*x^2*(k+1)*(2*k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 11 2013.
a(n) ~ 2^(3*n+5/4)*n^n/(exp(n)*Pi^(n+1/2)). - Vaclav Kotesovec, Jun 26 2013
a(n) = Sum_{k=0..n} A007696(k) * (4*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025
EXAMPLE
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 29*x^3/3! + 265*x^4/4! + 3001*x^5/5! +...
log(A(x)) = x + 4*x^2/2! + 16*x^3/3! + 128*x^4/4! + 1280*x^5/5! +...
A(x)^2/A(-x)^2 = 1 + 4*x + 16*x^2/2! + 128*x^3/3! +...+ 4^n*A000111(n)*x^n/n! +...
O.g.f.: 1/(1-x - 4*1*1*x^2/(1-5*x - 4*2*3*x^2/(1-9*x - 4*3*5*x^2/(1-13*x - 4*4*7*x^2/(1-17*x - 4*5*9*x^2/(1-...)))))) [continued fraction by Sergei Gladkovskii].
MATHEMATICA
CoefficientList[Series[1/(1-Sin[4*x])^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)
PROG
(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(2*X)-sin(2*X))^(-1/2), 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^2/subst(A^2, x, -x)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* From A'(x) = A(x)^3 / A(-x)^2: */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^3/subst(A, x, -x)^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* 1/sqrt(1-2*Series_Reversion(Integral 1/sqrt(1+4*x-4*x^2) dx)): */
{a(n)=local(A=1); A=1/sqrt(1-2*serreverse(intformal(1/sqrt(1+4*x-4*x^2 +x*O(x^n))))); 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));
a007696(n) = prod(k=0, n-1, 4*k+1);
a(n) = sum(k=0, n, a007696(k)*(4*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 09 2008
STATUS
approved