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 A325292 E.g.f. S(x), where C(x*y) + S(x*y) = exp( Integral Integral C(x*y) dx dy ) such that C(x)^2 - S(x)^2 = 1. 4

%I #16 Apr 15 2023 14:16:08

%S 1,8,576,160768,123535360,212713734144,716196297048064,

%T 4280584942657732608,42250703121584165486592,

%U 651154631135458759089848320,14983590319172065236171175755776,496301942561421311900528265903734784,22953613919171561374366988621726483480576,1444609513446024762131466039751756562435145728

%N E.g.f. S(x), where C(x*y) + S(x*y) = exp( Integral Integral C(x*y) dx dy ) such that C(x)^2 - S(x)^2 = 1.

%C Unsigned version of A326552.

%H Paul D. Hanna, <a href="/A325292/b325292.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. S(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)!^2, where series S(x) and related series C(x) satisfy the following relations.

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

%F (1.b) C'(x)/S(x) = S'(x)/C(x) = 1/x * Integral C(x) dx.

%F (2.a) S(x) = Integral C(x)/x * (Integral C(x) dx) dx.

%F (2.b) C(x) = 1 + Integral S(x)/x * (Integral C(x) dx) dx.

%F (3.a) C(x) + S(x) = exp( Integral 1/x * (Integral C(x) dx) dx ).

%F (3.b) C(x) = cosh( Integral 1/x * (Integral C(x) dx) dx ).

%F (3.c) S(x) = sinh( Integral 1/x * (Integral C(x) dx) dx ).

%F Integration.

%F (4.a) S(x*y) = Integral C(x*y) * (Integral C(x*y) dy) dx.

%F (4.b) C(x*y) = 1 + Integral S(x*y) * (Integral C(x*y) dy) dx.

%F (4.c) S(x*y) = Integral C(x*y) * (Integral C(x*y) dx) dy.

%F (4.d) C(x*y) = 1 + Integral S(x*y) * (Integral C(x*y) dx) dy.

%F Exponential.

%F (5.a) C(x*y) + S(x*y) = exp( Integral Integral C(x*y) dx dy ).

%F (5.b) C(x*y) = cosh( Integral Integral C(x*y) dx dy ).

%F (5.c) S(x*y) = sinh( Integral Integral C(x*y) dx dy ).

%F Derivatives.

%F (6.a) d/dx S(x*y) = C(x*y) * Integral C(x*y) dy.

%F (6.b) d/dx C(x*y) = S(x*y) * Integral C(x*y) dy.

%F (6.c) d/dy S(x*y) = C(x*y) * Integral C(x*y) dx.

%F (6.d) d/dy C(x*y) = S(x*y) * Integral C(x*y) dx.

%e E.g.f. S(x) = x + 8*x^3/3!^2 + 576*x^5/5!^2 + 160768*x^7/7!^2 + 123535360*x^9/9!^2 + 212713734144*x^11/11!^2 + 716196297048064*x^13/13!^2 + 4280584942657732608*x^15/15!^2 + 42250703121584165486592*x^17/17!^2 + 651154631135458759089848320*x^19/19!^2 + 14983590319172065236171175755776*x^21/21!^2 + ...

%e where S(x) = sinh( Integral 1/x * (Integral C(x) dx) dx ),

%e also, S(x*y) = sinh( Integral Integral C(x*y) dx dy ).

%e RELATED SERIES.

%e C(x) = 1 + 2*x^2/2!^2 + 56*x^4/4!^2 + 8336*x^6/6!^2 + 3985792*x^8/8!^2 + 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 + 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 + 4975999084909976839454720*x^18/18!^2 + 94178912073481319162642169856*x^20/20!^2 + ...

%e where C(x) = cosh( Integral 1/x * (Integral C(x) dx) dx ),

%e also, C(x*y) = cosh( Integral Integral C(x*y) dx dy ).

%e SPECIFIC VALUES.

%e At x = 1/2,

%e C(1/2) = 1.13133757946411922642102833324416139...

%e S(1/2) = 0.52907912329606456055608764850290077...

%e log(C(1/2) + S(1/2)) = 0.50706859662590456104854330721421537...

%e At x = 1,

%e C(1) = 1.61616724447561044622618032294959193...

%e S(1) = 1.26964426597212165112687564431552303...

%e log(C(1) + S(1)) = 1.05980614652360497313310791544203867...

%e At x = 2,

%e C(2) = 7.0181980831554020705059330009720760...

%e S(2) = 6.9465894030384550946994132182413166...

%e log(C(2) + S(2)) = 2.636538981679765615420983831302958...

%e At x = 3, the power series for C(x) and S(x) diverge.

%o (PARI) {a(n) = my(C=1, S=x); for(i=1, 2*n+1,

%o S = intformal( C/x * intformal( C +x*O(x^(2*n+1)) ) );

%o C = 1 + intformal( S/x * intformal( C +x*O(x^(2*n+1)) ) ); ); (2*n+1)!^2*polcoeff(S, 2*n+1)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A325290 (C+S), A325291 (C).

%Y Cf. A326552.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 16 2019

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