A215639 E.g.f. satisfies: A(x) = cos(x*A(x)) + sin(x*A(x)).

1, 1, 1, -4, -51, -304, 125, 34880, 557753, 3416320, -74779911, -2917151744, -46015368443, 115191402496, 30429734385973, 942941062774784, 9925460231059185, -471696770041053184, -29508689065235461903, -733077456673636089856, 4714209123766494329021

Offset: 0

Author

Paul D. Hanna, Aug 18 2012

Keywords

• sign

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! - 4*x^3/3! - 51*x^4/4! - 304*x^5/5! + 125*x^6/6! +...
where A(x) = cos(x*A(x)) + sin(x*A(x)).
Related expansions:
cos(x*A(x)) = 1 - x^2/2! - 6*x^3/3! - 23*x^4/4! + 40*x^5/5! + 2159*x^6/6! + 26656*x^7/7! + 114577*x^8/8! +...
sin(x*A(x)) = x + 2*x^2/2! + 2*x^3/3! - 28*x^4/4! - 344*x^5/5! - 2034*x^6/6! + 8224*x^7/7! + 443176*x^8/8! +...
		

Crossrefs

Cf. A215638, A201923.

Programs

PARI

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff(1/x*serreverse(x/(cos(X)+sin(X))), n)}

PARI

{a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cos(X)+sin(X))^(n+1)/(n+1), n)}

PARI

{a(n)=local(A=1+x+x^2*O(x^n)); for(i=1,n,A=cos(x*A)+sin(x*A));n!*polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Formula

E.g.f.: A(x) = (1/x)*Series_Reversion( x/(cos(x) + sin(x)) ).

E.g.f. satisfies: A(x/(cos(x) + sin(x))) = cos(x) + sin(x).

a(n) = [x^n/n!] (cos(x)+sin(x))^(n+1) / (n+1).

a(n) = A215638(n+1)/(n+1) for n>=0.