cp's OEIS Frontend

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

%I A290880 #11 Aug 27 2017 20:02:34
%S A290880 1,1,-7,265,-24175,4037425,-1070526775,412826556025,-218150106913375,
%T A290880 151297155973926625,-133288452772763494375,145378048431548466795625,
%U A290880 -192296944484564858674279375,303266384253858232005535140625,-562167814015907092875287424484375,1210147640238238850996978598797265625,-2993757681527630470101347134338702109375
%N A290880 E.g.f. C(x) satisfies: C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1, where S(x) is the e.g.f. of A290881.
%H A290880 Paul D. Hanna, <a href="/A290880/b290880.txt">Table of n, a(n) for n = 0..100</a>
%F A290880 E.g.f.: C(x) = cosh( Series_Reversion( Integral sqrt( cosh(2*x) ) dx ) ).
%F A290880 Let S(x) be the e.g.f. of A290881, then:
%F A290880 (1) C'(x) = S(x) / sqrt(C(x)^2 + S(x)^2).
%F A290880 (2) S'(x) = C(x) / sqrt(C(x)^2 + S(x)^2).
%F A290880 such that C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.
%e A290880 E.g.f.: 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! +...
%e A290880 such that C(x)^2 - S(x)^2 = 1 where S(x) begins:
%e A290880 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! +...
%o A290880 (PARI) {a(n) = my(C=1,S=x); for(i=1,n, C = 1 + intformal( S/sqrt(C^2 + S^2 + O(x^(2*n+2))) ); S = intformal( C/sqrt(C^2 + S^2)) ); (2*n)!*polcoeff(C,2*n)}
%o A290880 for(n=0,20, print1(a(n),", "))
%o A290880 (PARI) {a(n) = my(C=1); C = cosh( serreverse( intformal( sqrt(cosh(2*x + O(x^(2*n+2)))) ) )); (2*n)!*polcoeff(C,2*n)}
%o A290880 for(n=0,20, print1(a(n),", "))
%Y A290880 Cf. A290879, A290881, A290882, A290883, A153302.
%K A290880 sign
%O A290880 0,3
%A A290880 _Paul D. Hanna_, Aug 13 2017