A289695 E.g.f.: S(x) satisfies: S(x) = Integral sqrt( (1 + S(x)^2)*(1 + 2*S(x)^2) ) dx.

1, 3, 33, 819, 34209, 2189187, 198717057, 24278289651, 3842052205761, 764478393601923, 186805627856569953, 54994092004290217779, 19197418751181422089569, 7840711973025043515377667, 3704137338316764145483007937, 2004220869541285849551954747891

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..228

Formula

E.g.f.: S(x) = Series_Reversion( Integral 1/sqrt( (1 + x^2)*(1 + 2*x^2) ) dx ).

E.g.f.: S(x) = sinh( Series_Reversion( Integral 1/sqrt( cosh(2*x) ) dx ) ).

E.g.f.: S(x) = Integral sqrt(1 + S(i*x)^2) / (1 + 2*S(i*x)^2) dx, where i^2 = -1.

Let C(x) be the e.g.f. of A193543, then

(1) S(x) = sqrt(C(x)^2 - 1),

(2) S(x) = Integral C(x) * sqrt(C(x)^2 + S(x)^2) dx,

(3) C(x) = 1 + Integral S(x) * sqrt(C(x)^2 + S(x)^2) dx,

(4) C(x) + S(x) = exp( Integral sqrt(1 + 2*S(x)^2) ) dx ).

Example

E.g.f.: S(x) = x + 3*x^3/3! + 33*x^5/5! + 819*x^7/7! + 34209*x^9/9! + 2189187*x^11/11! + 198717057*x^13/13! + 24278289651*x^15/15! + 3842052205761*x^17/17! + 764478393601923*x^19/19! + 186805627856569953*x^21/21! +...
where
sqrt((1 + S(x)^2)*(1 + 2*S(x)^2)) = 1 + 3*x^2/2! + 33*x^4/4! + 819*x^6/6! + 34209*x^8/8! +...
RELATED SERIES.
C(x) = sqrt(1 + S(x)^2) equals the e.g.f. of A193543, and begins
C(x) = 1 + x^2/2! + 9*x^4/4! + 153*x^6/6! + 4977*x^8/8! + 261009*x^10/10! + 20039481*x^12/12! + 2121958377*x^14/14! + 296297348193*x^16/16! +...

Prog

(PARI) {a(n) = my(S=1); S = sinh( serreverse( intformal( 1/sqrt( cosh(2*x +O(x^(2*n+1))) ) ) ) ); (2*n-1)!*polcoeff(S, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A193543.

Sequence in context: A229513 A210833 A174488 * A371683 A124432 A234715

Adjacent sequences: A289692 A289693 A289694 * A289696 A289697 A289698

Keyword

nonn

Author

Paul D. Hanna, Aug 14 2017

Status

approved