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 A326799 Consider the e.g.f. C(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)! and related functions A(x,y) and B(x,y), as defined in the Formula section. Sequence gives the triangular array of coefficients T(n,k) (n>=0, 0<=k<=n) of C(x,y). 6
 2, 0, -4, 0, -160, 6, 0, 0, 1344, -8, 0, 0, 145152, -5760, 10, 0, 0, 0, -2534400, 17600, -12, 0, 0, 0, -551755776, 20500480, -43680, 14, 0, 0, 0, 0, 16400384000, -109025280, 94080, -16, 0, 0, 0, 0, 6006289203200, -213971337216, 441423360, -182784, 18 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The e.g.f. C(x,y) at y = x is described by A326796. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..1890 FORMULA The e.g.f. Cx = C(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)! and related functions Ax = A(x,y), Bx = B(x,y), Ay = A(y,x), By = B(y,x), and Cy = C(y,x) satisfy the following relations. DEFINITION. (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. IDENTITIES. (3a) Ax^2 + Bx^2 + Cx^2 = 1. (3b) Ay^2 + By^2 + Cy^2 = 1. (4a) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2 = 1. (4b) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2 = 1. (5a) Ax*(d/dx Ax) + Bx*(d/dx Bx) + Cx*(d/dx Cx) = 0. (5b) Ay*(d/dy Ay) + By*(d/dy By) + Cy*(d/dy Cy) = 0. (5c) Ax*(d/dy Ay) + Bx*(d/dy By) + Cx*(d/dy Cy) = 0. (5d) Ay*(d/dx Ax) + By*(d/dx Bx) + Cy*(d/dx Cx) = 0. (5e) Ax*(d/dy Ax) + Bx*(d/dy Bx) + Cx*(d/dy Cx) = 0. (5f) Ay*(d/dx Ay) + By*(d/dx By) + Cy*(d/dx Cy) = 0. RELATED FUNCTIONS. (6a) SS(x*y) = Ax*Ay + Bx*By + Cx*Cy. (6b) d/dx SS(x*y) = Ax*(d/dx Ay) + Bx*(d/dx By) + Cx*(d/dx Cy). (6c) d/dy SS(x*y) = Ay*(d/dy Ax) + By*(d/dy Bx) + Cy*(d/dy Cx). (7a) CC(x*y)^2 = (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2. (7b) CC(x*y)^2 = (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2. (7c) CC(x*y)^2 = (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2. In the above, CC(x) and SS(x) are the e.g.f.s of A326551 and A326552 defined by (8a) CC(x*y)^2 + SS(x*y)^2 = 1, (8b) SS(x*y) = Integral CC(x*y) * (Integral CC(x*y) dy) dx, (8c) CC(x*y) = 1 - Integral SS(x*y) * (Integral CC(x*y) dy) dx, (8d) SS(x*y) = sin( Integral Integral CC(x*y) dx dy ), (8e) CC(x*y) = cos( Integral Integral CC(x*y) dx dy ). OTHER RELATIONS. (9a) Ay = Ax*SS(x*y) - Bx*(d/dx Cx) + Cx*(d/dx Bx). (9b) By = Bx*SS(x*y) - Cx*(d/dx Ax) + Ax*(d/dx Cx). (9c) Cy = Cx*SS(x*y) - Ax*(d/dx Bx) + Bx*(d/dx Ax). (9d) Ax = Ay*SS(x*y) - By*(d/dy Cy) + Cy*(d/dy By). (9e) Bx = By*SS(x*y) - Cy*(d/dy Ay) + Ay*(d/dy Cy). (9f) Cx = Cy*SS(x*y) - Ay*(d/dy By) + By*(d/dy Ay). DERIVATIVES. (10a) d/dx Ax = Bx*Cy - Cx*By. (10b) d/dx Bx = Cx*Ay - Ax*Cy. (10c) d/dx Cx = Ax*By - Bx*Ay. (10d) d/dy Ay = By*Cx - Cy*Bx. (10e) d/dy By = Cy*Ax - Ay*Cx. (10f) d/dy Cy = Ay*Bx - By*Ax. EXAMPLE E.g.f.: C(x,y) = (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! + (-551755776*x^7*y^7 + 20500480*x^5*y^9 - 43680*x^3*y^11 + 14*x*y^13)/14! + (16400384000*x^7*y^9 - 109025280*x^5*y^11 + 94080*x^3*y^13 - 16*x*y^15)/16! + (6006289203200*x^9*y^9 - 213971337216*x^7*y^11 + 441423360*x^5*y^13 - 182784*x^3*y^15 + 18*x*y^17)/18! + (-271368838840320*x^9*y^11 + 1750895247360*x^7*y^13 - 1472507904*x^5*y^15 + 328320*x^3*y^17 - 20*x*y^19)/20! + (-150055074904670208*x^11*y^11 + 5196968265646080*x^9*y^13 - 10481366827008*x^7*y^15 + 4247147520*x^5*y^17 - 554400*x^3*y^19 + 22*x*y^21)/22! + ... such that . C(x,y) = 0 + Integral A(x,y)*B(y,x) - B(x,y)*A(y,x) dx, . C(y,x) = 1 + Integral A(y,x)*B(x,y) - B(y,x)*A(x,y) dy, where A(x,y) and B(x,y) satisfy . A(x,y)^2 + B(x,y)^2 + C(x,y)^2 = 1. TRIANGLE. This triangle of coefficients T(n,k) of x^(2*n-2*k+1)*y^(2*k+1)/(2*n+2)! in C(x,y) begins 2; 0, -4; 0, -160, 6; 0, 0, 1344, -8; 0, 0, 145152, -5760, 10; 0, 0, 0, -2534400, 17600, -12; 0, 0, 0, -551755776, 20500480, -43680, 14; 0, 0, 0, 0, 16400384000, -109025280, 94080, -16; 0, 0, 0, 0, 6006289203200, -213971337216, 441423360, -182784, 18; 0, 0, 0, 0, 0, -271368838840320, 1750895247360, -1472507904, 328320, -20; 0, 0, 0, 0, 0, -150055074904670208, 5196968265646080, -10481366827008, 4247147520, -554400, 22; 0, 0, 0, 0, 0, 0, 9574791098375602176, -60514176440205312, 49984638713856, -10935429120, 890560, -24; 0, 0, 0, 0, 0, 0, 7448871207078094438400, -252719117696793313280, 501704844897484800, -200297889792000, 25701561600, -1372800, 26; ... RELATED FUNCTIONS. A(x,y) = 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! + (1*x^13 - 7722*x^11*y^2 + 4279275*x^9*y^4 - 52144092*x^7*y^6 + 7702695*x^5*y^8 - 28314*x^3*y^10 + 13*x*y^12)/13! + (-1*x^15 + 15015*x^13*y^2 - 22387365*x^11*y^4 + 2701093395*x^9*y^6 + 3472834365*x^7*y^8 - 49252203*x^5*y^10 + 65065*x^3*y^12 - 15*x*y^14)/15! + ... such that . A(x,y) = 0 + Integral B(x,y)*C(y,x) - C(x,y)*B(y,x) dx, . A(y,x) = 0 + Integral B(y,x)*C(x,y) - C(y,x)*B(x,y) dy. B(x,y) = 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! + (-1*x^14 + 10920*x^12*y^2 - 10250240*x^10*y^4 + 482786304*x^8*y^6)/14! + (1*x^16 - 20160*x^14*y^2 + 45427200*x^12*y^4 - 11480268800*x^10*y^6)/16! + ... such that . B(x,y) = 1 + Integral C(x,y)*A(y,x) - A(x,y)*C(y,x) dx, . B(y,x) = 0 + Integral C(y,x)*A(x,y) - A(y,x)*C(x,y) dy. CC(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 + ... + A326551(n)*x^(2*n)/(2*n)!^2 + ... such that . CC(x*y) = sqrt( (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2 ). SS(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 + ... + A326552(n)*x^(2*n+1)/(2*n+1)!^2 + ... such that SS(x*y) = Ax*Ay + Bx*By + Cx*Cy. PROG (PARI) {TCx(n, k) = my(Ax=x, Bx=1, Cx=x, Ay=y, By=y, Cy=1); for(i=0, 2*n+1, Ax = 0 + intformal( Bx*Cy - Cx*By, x) + O(x^(2*n+2)); Bx = 1 + intformal( Cx*Ay - Ax*Cy, x) + O(x^(2*n+2)); Cx = 0 + intformal( Ax*By - Bx*Ay, x) + O(x^(2*n+2)); Ay = 0 + intformal( By*Cx - Cy*Bx, y) + O(y^(2*n+2)); By = 0 + intformal( Cy*Ax - Ay*Cx, y) + O(y^(2*n+2)); Cy = 1 + intformal( Ay*Bx - By*Ax, y) + O(y^(2*n+2)); ); (2*n+2)! * polcoeff( polcoeff(Cx, 2*n-2*k+1, x), 2*k+1, y)} for(n=0, 10, for(k=0, n, print1( TCx(n, k), ", ")); print("")) CROSSREFS Cf. A326797 (A), A326798 (B). Cf. A326796 (row sums), A326551 (CC), A326552 (SS). Sequence in context: A285774 A262589 A019215 * A112081 A265158 A366907 Adjacent sequences: A326796 A326797 A326798 * A326800 A326801 A326802 KEYWORD sign,tabl AUTHOR Paul D. Hanna, Aug 03 2019 STATUS approved

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Last modified May 22 19:06 EDT 2024. Contains 372758 sequences. (Running on oeis4.)