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%I #15 Aug 27 2017 20:01:41
%S 1,-1,25,-1705,227665,-50333425,16655398825,-7711225809625,
%T 4760499335502625,-3779764853639958625,3752942823715824285625,
%U -4556465805050372544735625,6641455313355871353308640625,-11445605320939175012746492140625,23021828780691053491298409381015625,-53450977127256739279274500814544765625
%N E.g.f. S(x) satisfies: 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.
%H Paul D. Hanna, <a href="/A290881/b290881.txt">Table of n, a(n) for n = 1..100</a>
%F E.g.f.: S(x) = Series_Reversion( Integral sqrt( (1 + 2*x^2) / (1 + x^2) ) dx ).
%F E.g.f.: S(x) = sinh( Series_Reversion( Integral sqrt( cosh(2*x) ) dx ) ).
%F Let C(x) be the e.g.f. of A290880, then:
%F (1) C'(x) = S(x) / sqrt(C(x)^2 + S(x)^2),
%F (2) S'(x) = C(x) / sqrt(C(x)^2 + S(x)^2),
%F such that C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.
%e E.g.f.: 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! +...
%e such that C(x)^2 - S(x)^2 = 1 where C(x) begins:
%e 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! +...
%o (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-1)!*polcoeff(S,2*n-1)}
%o for(n=1,20, print1(a(n),", "))
%o (PARI) {a(n) = my(C=1); S = serreverse( intformal( sqrt( (1+2*x^2) / (1+x^2 + O(x^(2*n+2))) ) )); (2*n-1)!*polcoeff(S,2*n-1)}
%o for(n=1,20, print1(a(n),", "))
%o (PARI) {a(n) = my(S=x); S = sinh( serreverse( intformal( sqrt(cosh(2*x + O(x^(2*n+2)))) ) )); (2*n-1)!*polcoeff(S,2*n-1)}
%o for(n=1,20, print1(a(n),", "))
%Y Cf. A290879, A290880, A290882, A290883, A153302.
%K sign
%O 1,3
%A _Paul D. Hanna_, Aug 13 2017
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