%I #19 Aug 24 2019 11:28:06
%S 1,-8,576,-160768,123535360,-212713734144,716196297048064,
%T -4280584942657732608,42250703121584165486592,
%U -651154631135458759089848320,14983590319172065236171175755776,-496301942561421311900528265903734784,22953613919171561374366988621726483480576,-1444609513446024762131466039751756562435145728,121022534222796916421149671221445519229890299166720
%N E.g.f. S(x), where C(x*y) + iS(x*y) = exp( i*Integral Integral C(x*y) dx dy ) such that C(x)^2 + S(x)^2 = 1.
%C The hyperbolic analog of the e.g.f. is described by A325292.
%C The e.g.f. can be derived from the functions described by A326797, A326798, and A326799.
%C The e.g.f. can be derived from the functions described by A326800, A326801, and A326802.
%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) S'(x)/C(x) = -C'(x)/S(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) + i*S(x) = exp( i*Integral 1/x * (Integral C(x) dx) dx ).
%F (3.b) C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ).
%F (3.c) S(x) = sin( 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) + i*S(x*y) = exp( i*Integral Integral C(x*y) dx dy ).
%F (5.b) C(x*y) = cos( Integral Integral C(x*y) dx dy ).
%F (5.c) S(x*y) = sin( 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) = sin( Integral 1/x * (Integral C(x) dx) dx ),
%e also, S(x*y) = sin( Integral Integral C(x*y) dx dy ),
%e such that C(x)^2 + S(x)^2 = 1.
%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) = cos( Integral 1/x * (Integral C(x) dx) dx ),
%e also, C(x*y) = cos( Integral Integral C(x*y) dx dy ).
%e RELATED FUNCTIONS.
%e Given functions Ax, Bx, Cx, Ay, By, and Cy defined by
%e (1a) Ax = 0 + Integral Bx*Cy - Cx*By dx,
%e (1b) Bx = 1 + Integral Cx*Ay - Ax*Cy dx,
%e (1c) Cx = 0 + Integral Ax*By - Bx*Ay dx,
%e (2a) Ay = 0 + Integral By*Cx - Cy*Bx dy,
%e (2b) By = 0 + Integral Cy*Ax - Ay*Cx dy,
%e (2c) Cy = 1 + Integral Ay*Bx - By*Ax dy,
%e then
%e S(x*y) = Ax*Ay + Bx*By + Cx*Cy.
%e These related series begin as follows.
%e Ax = x + (-1*x^3 - 3*x*y^2)/3! + (1*x^5 - 30*x^3*y^2 + 5*x*y^4)/5! + (-1*x^7 + 315*x^5*y^2 + 525*x^3*y^4 - 7*x*y^6)/7! + (1*x^9 - 1260*x^7*y^2 + 18270*x^5*y^4 - 2940*x^3*y^6 + 9*x*y^8)/9! + (-1*x^11 + 3465*x^9*y^2 - 496650*x^7*y^4 - 695310*x^5*y^6 + 10395*x^3*y^8 - 11*x*y^10)/11! + ... (A326797)
%e Bx = 1 + (-1*x^2)/2! + (1*x^4)/4! + (-1*x^6 + 120*x^4*y^2)/6! + (1*x^8 - 672*x^6*y^2)/8! + (-1*x^10 + 2160*x^8*y^2 - 120960*x^6*y^4)/10! + (1*x^12 - 5280*x^10*y^2 + 1584000*x^8*y^4)/12! + ... (A326798)
%e Cx = (2*x*y)/2! + (-4*x*y^3)/4! + (-160*x^3*y^3 + 6*x*y^5)/6! + (1344*x^3*y^5 - 8*x*y^7)/8! + (145152*x^5*y^5 - 5760*x^3*y^7 + 10*x*y^9)/10! + (-2534400*x^5*y^7 + 17600*x^3*y^9 - 12*x*y^11)/12! + ... (A326799)
%e Ay = -y + (3*x^2*y + 1*y^3)/3! + (-5*x^4*y + 30*x^2*y^3 + -1*y^5)/5! + (7*x^6*y + -525*x^4*y^3 + -315*x^2*y^5 + 1*y^7)/7! + (-9*x^8*y + 2940*x^6*y^3 + -18270*x^4*y^5 + 1260*x^2*y^7 + -1*y^9)/9! + (11*x^10*y + -10395*x^8*y^3 + 695310*x^6*y^5 + 496650*x^4*y^7 + -3465*x^2*y^9 + 1*y^11)/11! + ...
%e By = (2*x*y)/2! + (-4*x^3*y)/4! + (6*x^5*y + -160*x^3*y^3)/6! + (-8*x^7*y + 1344*x^5*y^3)/8! + (10*x^9*y + -5760*x^7*y^3 + 145152*x^5*y^5)/10! + (-12*x^11*y + 17600*x^9*y^3 + -2534400*x^7*y^5)/12! + ...
%e Cy = 1 + (-1*y^2)/2! + (1*y^4)/4! + (120*x^2*y^4 + -1*y^6)/6! + (-672*x^2*y^6 + 1*y^8)/8! + (-120960*x^4*y^6 + 2160*x^2*y^8 + -1*y^10)/10! + (1584000*x^4*y^8 + -5280*x^2*y^10 + 1*y^12)/12! + ...
%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. A326551, A325292.
%Y Cf. A326797, A326798, A326799.
%Y Cf. A326800, A326801, A326802.
%K sign
%O 1,2
%A _Paul D. Hanna_, Jul 25 2019