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A367382
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E.g.f. G(x,y) = Integral C(x,y)*S(y,x) dx such that C(x,y)^2 + S(x,y)^2 = 1 and S(y,x) = Integral C(x,y)*C(y,x) dy, as a triangle of coefficients T(n,k) read by rows.
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1, -2, -1, 8, 12, 1, -32, -136, -94, -1, 128, 1760, 2400, 824, 1, -512, -25728, -62096, -47600, -7386, -1, 2048, 398848, 1750400, 2213120, 1038616, 66436, 1, -8192, -6318080, -53428992, -103849600, -84201600, -24216888, -597878, -1, 32768, 100786176, 1735589888, 5135488000, 6207805440, 3427293408, 586155056, 5380848, 1
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OFFSET
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0,2
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COMMENTS
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See triangles A367380 and A367381, which describe the related functions S(x,y) and C(x,y), respectively.
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LINKS
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FORMULA
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E.g.f. G(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k+1)*y^(2*k+1)/((2*n-2*k+1)!*(2*k+1)!) along with the related functions Cx = C(x,y), Cy = C(y,x), Sx = S(x,y), Sy = S(y,x), Gx = G(x,y), Gy = G(y,x), satisfy the following relations.
(1.a) Cx^2 + Sx^2 = 1.
(1.b) Cy^2 + Sy^2 = 1.
(2.a) d/dx Sx = Cx * Cy.
(2.b) d/dx Cx = -Sx * Cy.
(2.c) d/dx Gx = Sy * Cx.
(2.d) d/dy Sy = Cy * Cx.
(2.e) d/dy Cy = -Sy * Cx.
(2.f) d/dy Gy = Sx * Cy.
(2.g) d/dx Sy = -Cy * Gy.
(2.h) d/dx Cy = Sy * Gy.
(2.i) d/dy Sx = -Cx * Gx.
(2.j) d/dy Cx = Sx * Gx.
(3.a) Sx = Integral Cx * Cy dx.
(3.b) Sy = Integral Cy * Cx dy.
(3.c) Cx = 1 - Integral Sx * Cy dx.
(3.d) Cy = 1 - Integral Sy * Cx dy.
(3.e) Gx = Integral Sy * Cx dx.
(3.f) Gy = Integral Sx * Cy dy.
(4.a) Sx = sin( Integral Cy dx ).
(4.b) Sy = sin( Integral Cx dy ).
(4.c) Cx = cos( Integral Cy dx ).
(4.d) Cy = cos( Integral Cx dy ).
(5.a) Sx = sin(x) - Integral Cx * Gx dy.
(5.b) Sy = sin(y) - Integral Cy * Gy dx.
(5.c) Cx = cos(x) + Integral Sx * Gx dy.
(5.d) Cy = cos(y) + Integral Sy * Gy dx.
(6.a) Integral Cy dx = x - Integral Gx dy.
(6.b) Integral Cx dy = y - Integral Gy dx.
(6.c) x + Integral (Cx - Gx) dy = y + Integral (Cy - Gy) dx.
(7.a) (Cx + i*Sx) = exp( i*Integral Cy dx ).
(7.b) (Cy + i*Sy) = exp( i*Integral Cx dy ).
(7.c) (Cx + i*Sx) = exp( i*x - i*Integral Gx dy ).
(7.d) (Cy + i*Sy) = exp( i*y - i*Integral Gy dx ).
(8.a) (Cx*Cy - Sx*Sy) = cos(y) - Integral (Sx*Cy + Cx*Sy)*(Cy - Gy) dx.
(8.b) (Cx*Cy - Sx*Sy) = cos(x) - Integral (Sx*Cy + Cx*Sy)*(Cx - Gx) dy.
(8.c) (Sx*Cy + Cx*Sy) = sin(y) + Integral (Cx*Cy - Sx*Sy)*(Cy - Gy) dx.
(8.d) (Sx*Cy + Cx*Sy) = sin(x) + Integral (Cx*Cy - Sx*Sy)*(Cx - Gx) dy.
(9.a) (Cx + i*Sx)*(Cy + i*Sy) = exp(i*y) + i*Integral (Cx + i*Sx)*(Cy + i*Sy)*(Cy - Gy) dx.
(9.b) (Cx + i*Sx)*(Cy + i*Sy) = exp(i*x) + i*Integral (Cx + i*Sx)*(Cy + i*Sy)*(Cx - Gx) dy.
(9.c) (Cx + i*Sx)*(Cy + i*Sy) = exp( i*y + i*Integral (Cy - Gy) dx ).
(9.d) (Cx + i*Sx)*(Cy + i*Sy) = exp( i*x + i*Integral (Cx - Gx) dy ).
(9.e) (Cx + i*Sx)*(Cy + i*Sy) = exp( i*(x + y) - i*(Integral Gx dy) - i*(Integral Gy dx) ).
(9.f) (Cx + i*Sx)*(Cy + i*Sy) = exp( i*(x + y) - i*Integral Integral (Sx*Cy + Cx*Sy) dx dy ).
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EXAMPLE
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E.g.f. G(x,y) = x*y - (2*x^3*y/3! + 1*x*y^3/3!) + (8*x^5*y/5! + 12*x^3*y^3/(3!*3!) + 1*x*y^5/5!) - (32*x^7*y/7! + 136*x^5*y^3/(5!*3!) + 94*x^3*y^5/(3!*5!) + 1*x*y^7/7!) + (128*x^9*y/9! + 1760*x^7*y^3/(7!*3!) + 2400*x^5*y^5/(5!*5!) + 824*x^3*y^7/(3!*7!) + 1*x*y^9/9!) - (512*x^11*y/11! + 25728*x^9*y^3/(9!*3!) + 62096*x^7*y^5/(7!*5!) + 47600*x^5*y^7/(5!*7!) + 7386*x^3*y^9/(3!*9!) + 1*x*y^11/11!) + (2048*x^13*y/13! + 398848*x^11*y^3/(11!*3!) + 1750400*x^9*y^5/(9!*5!) + 2213120*x^7*y^7/(7!*7!) + 1038616*x^5*y^9/(5!*9!) + 66436*x^3*y^11/(3!*11!) + 1*x*y^13/13!) - (8192*x^15*y/15! + 6318080*x^13*y^3/(13!*3!) + 53428992*x^11*y^5/(11!*5!) + 103849600*x^9*y^7/(9!*7!) + 84201600*x^7*y^9/(7!*9!) + 24216888*x^5*y^11/(5!*11!) + 597878*x^3*y^13/(3!*13!) + 1*x*y^15/15!) + ...
This triangle of coefficients T(n,k) of x^(2*n-2*k+1)*y^(2*k+1)/((2*n-2*k+1)!*(2*k+1)!) in G(x,y) starts
1;
-2, -1;
8, 12, 1;
-32, -136, -94, -1;
128, 1760, 2400, 824, 1;
-512, -25728, -62096, -47600, -7386, -1;
2048, 398848, 1750400, 2213120, 1038616, 66436, 1;
-8192, -6318080, -53428992, -103849600, -84201600, -24216888, -597878, -1;
32768, 100786176, 1735589888, 5135488000, 6207805440, 3427293408, 586155056, 5380848, 1;
-131072, -1611169792, -58877726720, -268962125824, -460536301312, -387334142976, -147664947312, -14448604000, -48427570, -1; ...
RELATED SERIES.
The related series C(x,y) = Integral S(x,y) * C(y,x) dx begins
C(x,y) = 1 - 1*x^2/2! + (1*x^4/4! + 2*x^2*y^2/(2!*2!)) - (1*x^6/6! + 12*x^4*y^2/(4!*2!) + 8*x^2*y^4/(2!*4!)) + (1*x^8/8! + 94*x^6*y^2/(6!*2!) + 136*x^4*y^4/(4!*4!) + 32*x^2*y^6/(2!*6!)) - (1*x^10/10! + 824*x^8*y^2/(8!*2!) + 2400*x^6*y^4/(6!*4!) + 1760*x^4*y^6/(4!*6!) + 128*x^2*y^8/(2!*8!)) + (1*x^12/12! + 7386*x^10*y^2/(10!*2!) + 47600*x^8*y^4/(8!*4!) + 62096*x^6*y^6/(6!*6!) + 25728*x^4*y^8/(4!*8!) + 512*x^2*y^10/(2!*10!)) - (1*x^14/14! + 66436*x^12*y^2/(12!*2!) + 1038616*x^10*y^4/(10!*4!) + 2213120*x^8*y^6/(8!*6!) + 1750400*x^6*y^8/(6!*8!) + 398848*x^4*y^10/(4!*10!) + 2048*x^2*y^12/(2!*12!)) + (1*x^16/16! + 597878*x^14*y^2/(14!*2!) + 24216888*x^12*y^4/(12!*4!) + 84201600*x^10*y^6/(10!*6!) + 103849600*x^8*y^8/(8!*8!) + 53428992*x^6*y^10/(6!*10!) + 6318080*x^4*y^12/(4!*12!) + 8192*x^2*y^14/(2!*14!)) + ...
which also may be defined by
C(x,y) = cos( Integral C(y,x) dx ), and
C(y,x) = cos( Integral C(x,y) dy ).
The related series S(x,y), where S(x,y)^2 = 1 - C(x,y)^2, begins
S(x,y) = 1*x - (1*x^3/3! + 1*x*y^2/(1!*2!)) + (1*x^5/5! + 5*x^3*y^2/(3!*2!) + 1*x*y^4/(1!*4!)) - (1*x^7/7! + 33*x^5*y^2/(5!*2!) + 33*x^3*y^4/(3!*4!) + 1*x*y^6/(1!*6!)) + (1*x^9/9! + 277*x^7*y^2/(7!*2!) + 561*x^5*y^4/(5!*4!) + 277*x^3*y^6/(3!*6!) + 1*x*y^8/(1!*8!)) - (1*x^11/11! + 2465*x^9*y^2/(9!*2!) + 10545*x^7*y^4/(7!*4!) + 10545*x^5*y^6/(5!*6!) + 2465*x^3*y^8/(3!*8!) + 1*x*y^10/(1!*10!)) + (1*x^13/13! + 22149*x^11*y^2/(11!*2!) + 220065*x^9*y^4/(9!*4!) + 368213*x^7*y^6/(7!*6!) + 220065*x^5*y^8/(5!*8!) + 22149*x^3*y^10/(3!*10!) + 1*x*y^12/(1!*12!)) - (1*x^15/15! + 199297*x^13*y^2/(13!*2!) + 4983681*x^11*y^4/(11!*4!) + 13530881*x^9*y^6/(9!*6!) + 13530881*x^7*y^8/(7!*8!) + 4983681*x^5*y^10/(5!*10!) + 199297*x^3*y^12/(3!*12!) + 1*x*y^14/(1!*14!)) + ...
which also may be defined by
S(x,y) = sin( Integral sqrt(1 - S(y,x)^2) dx ), and
S(y,x) = sin( Integral sqrt(1 - S(x,y)^2) dy ).
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PROG
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(PARI) {T(n, k) = my(Sx=x, Sy=y, Cx=1, Cy=1); for(i=1, 2*n,
Sx = intformal( Cx*Cy +x*O(x^(2*n)), x);
Cx = 1 - intformal( Sx*Cy, x);
Sy = intformal( Cy*Cx +y*O(y^(2*k)), y);
Cy = 1 - intformal( Sy*Cx, y));
Gx = intformal( Sy*Cx, x);
(2*n-2*k+1)!*(2*k+1)! *polcoeff(polcoeff(Gx, 2*n-2*k+1, x), 2*k+1, y)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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