A281183 E.g.f. C(x)^2 = cosh( Integral C(x)^3 dx )^2 where C(x) is described by A281181.

1, 2, 32, 1376, 114176, 15519488, 3132551168, 879422726144, 327670676455424, 156439068819587072, 93116847688811282432, 67602541384815095054336, 58796336342280763841970176, 60351125684887424790500999168, 72187248798124538021926003539968, 99529442030183464236437157900713984, 156697512616609083360755035696397287424

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

E.g.f. C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).

E.g.f. C(x)^2 = d/dx Series_Reversion( (2*x + sin(2*x))/4 ).

E.g.f. C(x)^2 where related series S(x) and C(x) satisfy:

(1.a) C(x)^2 - S(x)^2 = 1.

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

Integrals.

(2.a) S(x) = Integral C(x)^4 dx.

(2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.

Exponential.

(3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).

(3.b) C(x) = cosh( Integral C(x)^3 dx ).

(3.c) S(x) = sinh( Integral C(x)^3 dx ).

Derivatives.

(4.a) S'(x) = C(x)^4.

(4.b) C'(x) = C(x)^3*S(x).

(4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.

(4.d) (C(x)^4 - S(x)^4)' = 4*C(x)^4*S(x).

Explicit Solutions.

(5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).

(5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).

(5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).

(5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).

(5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).

Example

E.g.f.: C(x)^2 =  1 + 2*x^2/2! + 32*x^4/4! + 1376*x^6/6! + 114176*x^8/8! + 15519488*x^10/10! + 3132551168*x^12/12! + 879422726144*x^14/14! + 327670676455424*x^16/16! + 156439068819587072*x^18/18! +...
such that C(x)^2 = 1 + S(x)^2 and the series for C(x) and S(x) begin:
C(x) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 37369*x^8/8! + 4732249*x^10/10! + 901188997*x^12/12! + 240798388357*x^14/14! + 85948640603761*x^16/16! +...+ A281181(n)*x^(2*n)/(2*n)! +...
S(x) = S(x) = x + 4*x^3/3! + 88*x^5/5! + 4672*x^7/7! + 454144*x^9/9! + 70084096*x^11/11! + 15728822272*x^13/13! + 4836914249728*x^15/15! + 1952137912385536*x^17/17! +...+ A281180(n)*x^(2*n-1)/(2*n-1)! +...

Prog

(PARI) {a(n) = my(S=x, C=1); for(i=1, n, S = intformal( C^4 +x*O(x^(2*n))); C = 1 + intformal( S*C^3 ) ); (2*n)!*polcoeff(C^2, 2*n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ) ) */
{a(n) = my(C2=x); C2 = deriv( serreverse( intformal( cos(x +x*O(x^(2*n)))^2 ))); (2*n)!*polcoeff(C2, 2*n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A281180 (S), A281181 (C), A281182 (C+S), A281184 (C^3).

Sequence in context: A320419 A012140 A268557 * A012209 A295418 A172286

Adjacent sequences: A281180 A281181 A281182 * A281184 A281185 A281186

Keyword

nonn

Author

Paul D. Hanna, Jan 16 2017

Status

approved