A290882 E.g.f. E(x) = C(x) + S(x) such that C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where C(x) is the e.g.f. of A290880 and S(x) is the e.g.f. of A290881.

1, 1, 1, -1, -7, 25, 265, -1705, -24175, 227665, 4037425, -50333425, -1070526775, 16655398825, 412826556025, -7711225809625, -218150106913375, 4760499335502625, 151297155973926625, -3779764853639958625, -133288452772763494375, 3752942823715824285625, 145378048431548466795625, -4556465805050372544735625, -192296944484564858674279375, 6641455313355871353308640625

Offset: 0

Author

Paul D. Hanna, Aug 13 2017

Keywords

• sign

Examples

			E.g.f.: E(x) = 1 + x + x^2/2! - x^3/3! - 7*x^4/4! + 25*x^5/5! + 265*x^6/6! - 1705*x^7/7! - 24175*x^8/8! + 227665*x^9/9! + 4037425*x^10/10! - 50333425*x^11/11! - 1070526775*x^12/12! + 16655398825*x^13/13! + 412826556025*x^14/14! - 7711225809625*x^15/15! - 218150106913375*x^16/16! +...
such that E(x) = C(x) + S(x) where
S(x) = x - x^3/3! + 25*x^5/5! - 1705*x^7/7! + 227665*x^9/9! - 50333425*x^11/11! + 16655398825*x^13/13! - 7711225809625*x^15/15! + 4760499335502625*x^17/17! - 3779764853639958625*x^19/19! + 3752942823715824285625*x^21/21! +...
C(x) = 1 + x^2/2! - 7*x^4/4! + 265*x^6/6! - 24175*x^8/8! + 4037425*x^10/10! - 1070526775*x^12/12! + 412826556025*x^14/14! - 218150106913375*x^16/16! + 151297155973926625*x^18/18! - 133288452772763494375*x^20/20! +...
These series satisfy: C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.
		

Links

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

Crossrefs

Cf. A290879, A290880, A290881, A290883.

Programs

PARI

{a(n) = my(C=1,S=x); for(i=1,n, C = 1 + intformal( S/sqrt(C^2 + S^2 + O(x^(n+2))) ); S = intformal( C/sqrt(C^2 + S^2)) ); n!*polcoeff(C + S,n)}
for(n=0,30, print1(a(n),", "))

PARI

{a(n) = my(E=1); E = exp( serreverse( intformal( sqrt(cosh(2*x + O(x^(n+2)))) ) )); n!*polcoeff(E,n)}
for(n=0,30, print1(a(n),", "))

Formula

E.g.f.: E(x) = exp( Series_Reversion( Integral sqrt( cosh(2*x) ) dx ) ).