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A326552 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. 8
1, -8, 576, -160768, 123535360, -212713734144, 716196297048064, -4280584942657732608, 42250703121584165486592, -651154631135458759089848320, 14983590319172065236171175755776, -496301942561421311900528265903734784, 22953613919171561374366988621726483480576, -1444609513446024762131466039751756562435145728, 121022534222796916421149671221445519229890299166720 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The hyperbolic analog of the e.g.f. is described by A325292.
The e.g.f. can be derived from the functions described by A326797, A326798, and A326799.
The e.g.f. can be derived from the functions described by A326800, A326801, and A326802.
LINKS
FORMULA
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.
(1.a) C(x)^2 + S(x)^2 = 1.
(1.b) S'(x)/C(x) = -C'(x)/S(x) = 1/x * Integral C(x) dx.
(2.a) S(x) = Integral C(x)/x * (Integral C(x) dx) dx.
(2.b) C(x) = 1 - Integral S(x)/x * (Integral C(x) dx) dx.
(3.a) C(x) + i*S(x) = exp( i*Integral 1/x * (Integral C(x) dx) dx ).
(3.b) C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ).
(3.c) S(x) = sin( Integral 1/x * (Integral C(x) dx) dx ).
Integration.
(4.a) S(x*y) = Integral C(x*y) * (Integral C(x*y) dy) dx.
(4.b) C(x*y) = 1 - Integral S(x*y) * (Integral C(x*y) dy) dx.
(4.c) S(x*y) = Integral C(x*y) * (Integral C(x*y) dx) dy.
(4.d) C(x*y) = 1 - Integral S(x*y) * (Integral C(x*y) dx) dy.
Exponential.
(5.a) C(x*y) + i*S(x*y) = exp( i*Integral Integral C(x*y) dx dy ).
(5.b) C(x*y) = cos( Integral Integral C(x*y) dx dy ).
(5.c) S(x*y) = sin( Integral Integral C(x*y) dx dy ).
Derivatives.
(6.a) d/dx S(x*y) = C(x*y) * Integral C(x*y) dy.
(6.b) d/dx C(x*y) = -S(x*y) * Integral C(x*y) dy.
(6.c) d/dy S(x*y) = C(x*y) * Integral C(x*y) dx.
(6.d) d/dy C(x*y) = -S(x*y) * Integral C(x*y) dx.
EXAMPLE
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 -+ ...
where S(x) = sin( Integral 1/x * (Integral C(x) dx) dx ),
also, S(x*y) = sin( Integral Integral C(x*y) dx dy ),
such that C(x)^2 + S(x)^2 = 1.
RELATED SERIES.
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 -+ ...
where C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ),
also, C(x*y) = cos( Integral Integral C(x*y) dx dy ).
RELATED FUNCTIONS.
Given functions Ax, Bx, Cx, Ay, By, and Cy defined by
(1a) Ax = 0 + Integral Bx*Cy - Cx*By dx,
(1b) Bx = 1 + Integral Cx*Ay - Ax*Cy dx,
(1c) Cx = 0 + Integral Ax*By - Bx*Ay dx,
(2a) Ay = 0 + Integral By*Cx - Cy*Bx dy,
(2b) By = 0 + Integral Cy*Ax - Ay*Cx dy,
(2c) Cy = 1 + Integral Ay*Bx - By*Ax dy,
then
S(x*y) = Ax*Ay + Bx*By + Cx*Cy.
These related series begin as follows.
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)
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)
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)
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! + ...
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! + ...
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! + ...
PROG
(PARI) {a(n) = my(C=1, S=x); for(i=1, 2*n+1,
S = intformal( C/x * intformal( C +x*O(x^(2*n+1)) ) );
C = 1 - intformal( S/x * intformal( C +x*O(x^(2*n+1)) ) ); ); (2*n+1)!^2*polcoeff(S, 2*n+1)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Sequence in context: A086641 A248706 A325292 * A303933 A188780 A015023
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 25 2019
STATUS
approved

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Last modified June 23 01:58 EDT 2024. Contains 373629 sequences. (Running on oeis4.)