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%I #35 Apr 15 2023 03:37:45
%S 1,-1,-3,1,-30,5,-1,315,525,-7,1,-1260,18270,-2940,9,-1,3465,-496650,
%T -695310,10395,-11,1,-7722,4279275,-52144092,7702695,-28314,13,-1,
%U 15015,-22387365,2701093395,3472834365,-49252203,65065,-15,1,-26520,86786700,-40541436936,454101489270,-63707972328,225645420,-132600,17
%N Consider the e.g.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k+1) * y^(2*k) / (2*n+1)! and related functions B(x,y) and C(x,y), as defined in the Formula section. Sequence gives the triangular array of coefficients T(n,k) (n>=0, 0<=k<=n) of A(x,y).
%C The e.g.f. of this triangle is equivalent to the e.g.f. of triangle A326800, where T(n,k) = A326800(n,k) * binomial(2*n+1, 2*k).
%C The e.g.f. A(x,y) at y = x is described by A326794.
%H Paul D. Hanna, <a href="/A326797/b326797.txt">Table of n, a(n) for n = 0..1890</a>
%F The e.g.f. Ax = A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^(2*n-2*k+1)*y^(2*k)/(2*n+1)! and related functions Bx = B(x,y), Cx = C(x,y), Ay = A(y,x), By = B(y,x), and Cy = C(y,x) satisfy the following relations.
%F DEFINITION.
%F (1a) Ax = 0 + Integral Bx*Cy - Cx*By dx,
%F (1b) Bx = 1 + Integral Cx*Ay - Ax*Cy dx,
%F (1c) Cx = 0 + Integral Ax*By - Bx*Ay dx.
%F (2a) Ay = 0 + Integral By*Cx - Cy*Bx dy,
%F (2b) By = 0 + Integral Cy*Ax - Ay*Cx dy,
%F (2c) Cy = 1 + Integral Ay*Bx - By*Ax dy.
%F IDENTITIES.
%F (3a) Ax^2 + Bx^2 + Cx^2 = 1.
%F (3b) Ay^2 + By^2 + Cy^2 = 1.
%F (4a) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2 = 1.
%F (4b) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2 = 1.
%F (5a) Ax*(d/dx Ax) + Bx*(d/dx Bx) + Cx*(d/dx Cx) = 0.
%F (5b) Ay*(d/dy Ay) + By*(d/dy By) + Cy*(d/dy Cy) = 0.
%F (5c) Ax*(d/dy Ay) + Bx*(d/dy By) + Cx*(d/dy Cy) = 0.
%F (5d) Ay*(d/dx Ax) + By*(d/dx Bx) + Cy*(d/dx Cx) = 0.
%F (5e) Ax*(d/dy Ax) + Bx*(d/dy Bx) + Cx*(d/dy Cx) = 0.
%F (5f) Ay*(d/dx Ay) + By*(d/dx By) + Cy*(d/dx Cy) = 0.
%F RELATED FUNCTIONS.
%F (6a) SS(x*y) = Ax*Ay + Bx*By + Cx*Cy.
%F (6b) d/dx SS(x*y) = Ax*(d/dx Ay) + Bx*(d/dx By) + Cx*(d/dx Cy).
%F (6c) d/dy SS(x*y) = Ay*(d/dy Ax) + By*(d/dy Bx) + Cy*(d/dy Cx).
%F (7a) CC(x*y)^2 = (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2.
%F (7b) CC(x*y)^2 = (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2.
%F (7c) CC(x*y)^2 = (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2.
%F In the above, CC(x) and SS(x) are the e.g.f.s of A326551 and A326552 defined by
%F (8a) CC(x*y)^2 + SS(x*y)^2 = 1,
%F (8b) SS(x*y) = Integral CC(x*y) * (Integral CC(x*y) dy) dx,
%F (8c) CC(x*y) = 1 - Integral SS(x*y) * (Integral CC(x*y) dy) dx,
%F (8d) SS(x*y) = sin( Integral Integral CC(x*y) dx dy ),
%F (8e) CC(x*y) = cos( Integral Integral CC(x*y) dx dy ).
%F OTHER RELATIONS.
%F (9a) Ay = Ax*SS(x*y) - Bx*(d/dx Cx) + Cx*(d/dx Bx).
%F (9b) By = Bx*SS(x*y) - Cx*(d/dx Ax) + Ax*(d/dx Cx).
%F (9c) Cy = Cx*SS(x*y) - Ax*(d/dx Bx) + Bx*(d/dx Ax).
%F (9d) Ax = Ay*SS(x*y) - By*(d/dy Cy) + Cy*(d/dy By).
%F (9e) Bx = By*SS(x*y) - Cy*(d/dy Ay) + Ay*(d/dy Cy).
%F (9f) Cx = Cy*SS(x*y) - Ay*(d/dy By) + By*(d/dy Ay).
%F DERIVATIVES.
%F (10a) d/dx Ax = Bx*Cy - Cx*By.
%F (10b) d/dx Bx = Cx*Ay - Ax*Cy.
%F (10c) d/dx Cx = Ax*By - Bx*Ay.
%F (10d) d/dy Ay = By*Cx - Cy*Bx.
%F (10e) d/dy By = Cy*Ax - Ay*Cx.
%F (10f) d/dy Cy = Ay*Bx - By*Ax.
%F VECTOR FORM.
%F Set radial vectors Vx = [Ax,Bx,Cx] and Vy = [Ay,By,Cy], then we can write the above relations in compact form using cross (X) products and dot (*) products.
%F (1) Vx = [0,1,0] + Integral Vx X Vy dx.
%F (2) Vy = [0,0,1] + Integral Vy X Vx dy.
%F (3a) Vx * Vx = 1.
%F (3b) Vy * Vy = 1.
%F (4a) (Vx * Vy)^2 + (d/dx Vx) * (d/dx Vx) = 1.
%F (4b) (Vx * Vy)^2 + (d/dy Vy) * (d/dy Vy) = 1.
%F (5a) Vx * (d/dx Vx) = 0.
%F (5b) Vy * (d/dy Vy) = 0.
%F (5c) Vx * (d/dy Vy) = 0.
%F (5d) Vy * (d/dx Vx) = 0.
%F (5e) Vx * (d/dy Vx) = 0.
%F (5f) Vy * (d/dx Vy) = 0.
%F (6a) SS(x*y) = Vx * Vy.
%F (6b) d/dx SS(x*y) = Vx * (d/dx Vy).
%F (6c) d/dy SS(x*y) = Vy * (d/dy Vx).
%F (7) CC(x*y)^2 = (Vx X Vy) * (Vx X Vy) = 1 - (Vx * Vy)^2.
%F (9a-c) Vy = Vx*SS(x*y) - Vx X (d/dx Vx) because Vx X (Vx X Vy) = Vx*(Vx * Vy) - Vy.
%F (9d-f) Vx = Vy*SS(x*y) - Vy X (d/dy Vy) because Vy X (Vy X Vx) = Vy*(Vx * Vy) - Vx.
%F (10a-c) d/dx Vx = Vx X Vy.
%F (10d-f) d/dy Vy = Vy X Vx.
%e E.g.f.: 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! + (1*x^17 - 26520*x^15*y^2 + 86786700*x^13*y^4 - 40541436936*x^11*y^6 + 454101489270*x^9*y^8 - 63707972328*x^7*y^10 + 225645420*x^5*y^12 - 132600*x^3*y^14 + 17*x*y^16)/17! +(-1*x^19 + 43605*x^17*y^2 - 274362660*x^15*y^4 + 345219726852*x^13*y^6 - 38308692031038*x^11*y^8 - 46821734704602*x^9*y^10 + 641122349868*x^7*y^12 - 823087980*x^5*y^14 + 247095*x^3*y^16 - 19*x*y^18)/19! + ...
%e such that
%e . A(x,y) = 0 + Integral B(x,y)*C(y,x) - C(x,y)*B(y,x) dx,
%e . A(y,x) = 0 + Integral B(y,x)*C(x,y) - C(y,x)*B(x,y) dy,
%e where B(x,y) and C(x,y) satisfy
%e . A(x,y)^2 + B(x,y)^2 + C(x,y)^2 = 1.
%e TRIANGLE.
%e This triangle of coefficients T(n,k) of x^(2*n-2*k+1) * y^(2*k) / (2*n+1)! in A(x,y) begins
%e 1;
%e -1, -3;
%e 1, -30, 5;
%e -1, 315, 525, -7;
%e 1, -1260, 18270, -2940, 9;
%e -1, 3465, -496650, -695310, 10395, -11;
%e 1, -7722, 4279275, -52144092, 7702695, -28314, 13;
%e -1, 15015, -22387365, 2701093395, 3472834365, -49252203, 65065, -15;
%e 1, -26520, 86786700, -40541436936, 454101489270, -63707972328, 225645420, -132600, 17;
%e -1, 43605, -274362660, 345219726852, -38308692031038, -46821734704602, 641122349868, -823087980, 247095, -19;
%e 1, -67830, 747317025, -2065684781160, 887850774580770, -9453938937390948, 1282451118838890, -4426467388200, 2540877885, -429590, 21; ...
%e RELATED TRIANGLE.
%e A related triangle (A326800), formed from coefficients of x^(2*n-2*k+1) * y^(2*k) / ((2*n-2*k+1)!*(2*k)!) in e.g.f. A(x,y), begins
%e 1;
%e -1, -1;
%e 1, -3, 1;
%e -1, 15, 15, -1;
%e 1, -35, 145, -35, 1;
%e -1, 63, -1505, -1505, 63, -1;
%e 1, -99, 5985, -30387, 5985, -99, 1;
%e -1, 143, -16401, 539679, 539679, -16401, 143, -1;
%e 1, -195, 36465, -3275811, 18679617, -3275811, 36465, -195, 1; ...
%e RELATED FUNCTIONS.
%e 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! + ...
%e such that
%e . B(x,y) = 1 + Integral C(x,y)*A(y,x) - A(x,y)*C(y,x) dx,
%e . B(y,x) = 0 + Integral C(y,x)*A(x,y) - A(y,x)*C(x,y) dy.
%e 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! + ...
%e such that
%e . C(x,y) = 0 + Integral A(x,y)*B(y,x) - B(x,y)*A(y,x) dx,
%e . C(y,x) = 1 + Integral A(y,x)*B(x,y) - B(y,x)*A(x,y) dy.
%e 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 + ...
%e such that
%e . CC(x*y) = sqrt( (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2 ).
%e 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 + ...
%e such that SS(x*y) = Ax*Ay + Bx*By + Cx*Cy.
%o (PARI) {TAx(n, k) = my(Ax=1, Bx=x, Cx=1, Ay=1, By=y, Cy=1);
%o for(i=0, 2*n+1,
%o Ax = 0 + intformal( Bx*Cy - Cx*By, x) + O(x^(2*n+2));
%o Bx = 1 + intformal( Cx*Ay - Ax*Cy, x) + O(x^(2*n+2));
%o Cx = 0 + intformal( Ax*By - Bx*Ay, x) + O(x^(2*n+2));
%o Ay = 0 + intformal( By*Cx - Cy*Bx, y) + O(y^(2*n+2));
%o By = 0 + intformal( Cy*Ax - Ay*Cx, y) + O(y^(2*n+2));
%o Cy = 1 + intformal( Ay*Bx - By*Ax, y) + O(y^(2*n+2));
%o );
%o (2*n+1)! * polcoeff( polcoeff(Ax, 2*n-2*k+1, x), 2*k, y)}
%o for(n=0, 10, for(k=0, n, print1( TAx(n, k), ", ")); print(""))
%Y Cf. A326798 (B), A326799 (C), A326800.
%Y Cf. A326794 (row sums), A326551 (CC), A326552 (SS).
%K sign,tabl,look
%O 0,3
%A _Paul D. Hanna_, Aug 03 2019
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