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 A367380 E.g.f. S(x,y) = Integral C(x,y)*C(y,x) dx such that C(x,y)^2 + S(x,y)^2 = 1 and C(y,x) = 1 - Integral S(y,x)*C(x,y) dy, as a triangle of coefficients T(n,k) read by rows. 4
 1, -1, -1, 1, 5, 1, -1, -33, -33, -1, 1, 277, 561, 277, 1, -1, -2465, -10545, -10545, -2465, -1, 1, 22149, 220065, 368213, 220065, 22149, 1, -1, -199297, -4983681, -13530881, -13530881, -4983681, -199297, -1, 1, 1793621, 118758993, 532981813, 799527361, 532981813, 118758993, 1793621, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS This triangle is a signed version of triangle A322220. See A367381 for the coefficients in the related function C(x,y). LINKS Paul D. Hanna, Table of n, a(n) for n = 0..1325 FORMULA E.g.f. S(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n+1-2*k)*y^(2*k)/((2*n+1-2*k)!*(2*k)!) 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 ). EXAMPLE E.g.f. S(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n+1-2*k)*y^(2*k)/((2*n+1-2*k)!*(2*k)!) 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!)) + ... The series S(x,y) may be defined by S(x,y) = Integral C(x,y) * C(y,x) dx such that C(x,y)^2 + S(x,y)^2 = 1. This triangle of coefficients T(n,k) of x^(2*n+1-2*k)*y^(2*k)/((2*n+1-2*k)!*(2*k)!) in S(x,y) begins 1; -1, -1; 1, 5, 1; -1, -33, -33, -1; 1, 277, 561, 277, 1; -1, -2465, -10545, -10545, -2465, -1; 1, 22149, 220065, 368213, 220065, 22149, 1; -1, -199297, -4983681, -13530881, -13530881, -4983681, -199297, -1; 1, 1793621, 118758993, 532981813, 799527361, 532981813, 118758993, 1793621, 1; -1, -16142529, -2905863441, -22355025777, -48737171169, -48737171169, -22355025777, -2905863441, -16142529, -1; ... RELATED SERIES. The related series C(x,y) (cf. A367381), where C(x,y)^2 + S(x,y)^2 = 1, 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!)) + ... This series 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 G(x,y) = Integral S(y,x) * C(x,y) dx begins 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!) + ... SPECIFIC VALUES. At x = Pi/4, y = Pi/4, C(Pi/4, Pi/4) = 0.831178816418434556618973541664847... S(Pi/4, Pi/4) = 0.556005193444494963502301287189184... G(Pi/4, Pi/4) = 0.490474107047544861372739409702074... At x = Pi/4, y = Pi/6, C(Pi/4, Pi/6) = 0.766334291094841251674725215860284... S(Pi/4, Pi/6) = 0.642442024070785014731997355171410... G(Pi/4, Pi/6) = 0.333312489674294244291205551151978... At x = Pi/6, y = Pi/4, C(Pi/6, Pi/4) = 0.928369578819309175865409938606631... S(Pi/6, Pi/4) = 0.371658344616205930317584779936615... G(Pi/6, Pi/4) = 0.350898319308070232265263489044750... At x = Pi/6, y = Pi/6, C(Pi/6, Pi/6) = 0.896526104547870943845048541964839... S(Pi/6, Pi/6) = 0.442990907202642228385926280567973... G(Pi/6, Pi/6) = 0.243469718547034805738161907279448... At x = Pi/8, y = Pi/8, C(Pi/8, Pi/8) = 0.934405560568629768616893625883648... S(Pi/8, Pi/8) = 0.356210960497322099066032510783456... G(Pi/8, Pi/8) = 0.143654626626560254692166383052725... PROG (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)); (2*n+1-2*k)!*(2*k)! *polcoeff(polcoeff(Sx, 2*n+1-2*k, x), 2*k, y)} for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print("")) CROSSREFS Cf. A367381 (C(x,y)), A322220. Sequence in context: A156587 A058720 A015116 * A322220 A174790 A156691 Adjacent sequences: A367377 A367378 A367379 * A367381 A367382 A367383 KEYWORD sign,tabl AUTHOR Paul D. Hanna, Nov 15 2023 STATUS approved

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