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%I #20 Apr 15 2023 03:17:24
%S 1,-1,-1,1,-3,1,-1,15,15,-1,1,-35,145,-35,1,-1,63,-1505,-1505,63,-1,1,
%T -99,5985,-30387,5985,-99,1,-1,143,-16401,539679,539679,-16401,143,-1,
%U 1,-195,36465,-3275811,18679617,-3275811,36465,-195,1,-1,255,-70785,12723711,-506849409,-506849409,12723711,-70785,255,-1
%N Consider the e.g.f. S(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k+1) * y^(2*k) / ((2*n-2*k+1)!*(2*k)!) and related functions C(x,y) and D(x,y), as defined in the Formula section. Sequence gives the triangular array of coefficients T(n,k) (n>=0, 0<=k<=n) of S(x,y).
%C The e.g.f. S(x,y) is equivalent to the e.g.f. of A326797.
%H Paul D. Hanna, <a href="/A326800/b326800.txt">Table of n, a(n) for n = 1..1891</a> (first 61 rows of this triangle).
%F The e.g.f. Sx = S(x,y) and related functions Cx = C(x,y), Dx = D(x,y), Sy = S(y,x), Cy = C(y,x), and Dy = D(y,x) satisfy the following relations.
%F DEFINITION.
%F (1a) Sx = Integral Cx*Dy + Cy*Dx dx,
%F (1b) Cx = sqrt(1/2) - Integral Sx*Dy + Sy*Dx dx,
%F (1c) Dx = sqrt(1/2) - Integral Sx*Cy - Sy*Cx dx,
%F (2a) Sy = Integral Cy*Dx + Cx*Dy dy,
%F (2b) Cy = sqrt(1/2) - Integral Sy*Dx + Sx*Dy dy,
%F (2c) Dy = sqrt(1/2) - Integral Sy*Cx - Sx*Cy dy.
%F IDENTITIES.
%F (3a) Dx^2 + Cx^2 + Sx^2 = 1.
%F (3b) Dy^2 + Cy^2 + Sy^2 = 1.
%F (4a) Dx*(d/dx Dx) + Cx*(d/dx Cx) + Sx*(d/dx Sx) = 0.
%F (4b) Dy*(d/dy Dy) + Cy*(d/dy Cy) + Sy*(d/dy Sy) = 0.
%F (4c) Dy*(d/dx Dx) - Cy*(d/dx Cx) - Sy*(d/dx Sx) = 0.
%F (4d) Dx*(d/dy Dy) - Cx*(d/dy Cy) - Sx*(d/dy Sy) = 0.
%F (5a) (Dx*Dy - Cx*Cy - Sx*Sy)^2 + (d/dx Dx)^2 + (d/dx Cx)^2 + (d/dx Sx)^2 = 1.
%F (5b) (Dx*Dy - Cx*Cy - Sx*Sy)^2 + (d/dy Dy)^2 + (d/dy Cy)^2 + (d/dy Sy)^2 = 1.
%F RELATED FUNCTIONS.
%F (6a) SS(x*y) = Dx*Dy - Cx*Cy - Sx*Sy.
%F (6b) d/dx SS(x*y) = Dx*(d/dx Dy) - Cx*(d/dx Cy) - Sx*(d/dx Sy).
%F (6c) d/dy SS(x*y) = Dy*(d/dy Dx) - Cy*(d/dy Cx) - Sy*(d/dy Sx).
%F (7a) CC(x*y)^2 = (Cx*Dy + Cy*Dx)^2 + (Sx*Dy + Sy*Dx)^2 + (Sx*Cy - Sy*Cx)^2.
%F (7b) CC(x*y)^2 = (d/dx Dx)^2 + (d/dx Cx)^2 + (d/dx Sx)^2.
%F (7c) CC(x*y)^2 = (d/dy Dy)^2 + (d/dy Cy)^2 + (d/dy Sy)^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 DERIVATIVES.
%F (9a) d/dx Sx = Cx*Dy + Cy*Dx.
%F (9b) d/dx Cx = -Sx*Dy - Sy*Dx.
%F (9c) d/dx Dx = -Sx*Cy + Sy*Cx.
%F (9d) d/dy Sy = Sy*Dx + Sx*Dy.
%F (9e) d/dy Cy = -Sy*Dx - Sx*Dy.
%F (9f) d/dy Dy = -Sy*Cx + Sx*Cy.
%e E.g.f.: S(x,y) = x + (-x^3/3! - x*y^2/2! ) + ( x^5/5! - 3*x^3*y^2/(3!*2!) + x*y^4/4! ) + (-x^7/7! + 15*x^5*y^2/(5!*2!) + 15*x^3*y^4/(3!*4!) - x*y^6/6! ) + ( x^9/9! - 35*x^7*y^2/(7!*2!) + 145*x^5*y^4/(5!*4!) - 35*x^3*y^6/(3!*6!) + x*y^8/8! ) + (-x^11/11! + 63*x^9*y^2/(9!*2!) - 1505*x^7*y^4/(7!*4!) - 1505*x^5*y^6/(5!*6!) + 63*x^3*y^8/(3!*8!) - x*y^10/10! ) + ( x^13/13! - 99*x^11*y^2/(11!*2!) + 5985*x^9*y^4/(9!*4!) - 30387*x^7*y^6/(7!*6!) + 5985*x^5*y^8/(5!*8!) - 99*x^3*y^10/(3!*10!) + x*y^12/12! ) + (-x^15/15! + 143*x^13*y^2/(13!*2!) - 16401*x^11*y^4/(11!*4!) + 539679*x^9*y^6/(9!*6!) + 539679*x^7*y^8/(7!*8!) - 16401*x^5*y^10/(5!*10!) + 143*x^3*y^12/(3!*12!) - x*y^14/14! ) + ( x^17/17! - 195*x^15*y^2/(15!*2!) + 36465*x^13*y^4/(13!*4!) - 3275811*x^11*y^6/(11!*6!) + 18679617*x^9*y^8/(9!*8!) - 3275811*x^7*y^10/(7!*10!) + 36465*x^5*y^12/(5!*12!) - 195*x^3*y^14/(3!*14!) + x*y^16/16! ) + (-x^19/19! + 255*x^17*y^2/(17!*2!) - 70785*x^15*y^4/(15!*4!) + 12723711*x^13*y^6/(13!*6!) - 506849409*x^11*y^8/(11!*8!) - 506849409*x^9*y^10/(9!*10!) + 12723711*x^7*y^12/(7!*12!) - 70785*x^5*y^14/(5!*14!) + 255*x^3*y^16/(3!*16!) - x*y^18/18! ) + ( x^21/21! - 323*x^19*y^2/(19!*2!) + 124865*x^17*y^4/(17!*4!) - 38067315*x^15*y^6/(15!*6!) + 4363117473*x^13*y^8/(13!*8!) - 26803260803*x^11*y^10/(11!*10!) + 4363117473*x^9*y^12/(9!*12!) - 38067315*x^7*y^14/(7!*14!) + 124865*x^5*y^16/(5!*16!) - 323*x^3*y^18/(3!*18!) + x*y^20/20! ) + ...
%e This triangle of coefficients T(n,k) of x^(2*n-2*k+1)*y^(2*k)/((2*n-2*k+1)!*(2*k)!) in e.g.f. S(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 -1, 255, -70785, 12723711, -506849409, -506849409, 12723711, -70785, 255, -1;
%e 1, -323, 124865, -38067315, 4363117473, -26803260803, 4363117473, -38067315, 124865, -323, 1;
%e -1, 399, -205105, 95686591, -22813329825, 1031421316783, 1031421316783, -22813329825, 95686591, -205105, 399, -1;
%e 1, -483, 318801, -212188067, 88405315713, -11952302851203, 77353020714385, -11952302851203, 88405315713, -212188067, 318801, -483, 1; ...
%e RELATED SERIES.
%e The e.g.f. of A326801 begins
%e C(x,y) = sqrt(1/2) * (1 + (-x^2/2! - x*y ) + ( x^4/4! + x*y^3/3! ) + (-x^6/6! + 8*x^4*y^2/(4!*2!) + 8*x^3*y^3/(3!*3!) - x*y^5/5! ) + ( x^8/8! - 24*x^6*y^2/(6!*2!) - 24*x^3*y^5/(3!*5!) + x*y^7/7! ) + (-x^10/10! + 48*x^8*y^2/(8!*2!) - 576*x^6*y^4/(6!*4!) - 576*x^5*y^5/(5!*5!) + 48*x^3*y^7/(3!*7!) - x*y^9/9! ) + ( x^12/12! - 80*x^10*y^2/(10!*2!) + 3200*x^8*y^4/(8!*4!) + 3200*x^5*y^7/(5!*7!) - 80*x^3*y^9/(3!*9!) + x*y^11/11! ) + (-x^14/14! + 120*x^12*y^2/(12!*2!) - 10240*x^10*y^4/(10!*4!) + 160768*x^8*y^6/(8!*6!) + 160768*x^7*y^7/(7!*7!) - 10240*x^5*y^9/(5!*9!) + 120*x^3*y^11/(3!*11!) - x*y^13/13! ) + ...).
%e The e.g.f. of A326802 begins
%e D(x,y) = sqrt(1/2) * (1 + (-x^2/2! + x*y ) + ( x^4/4! - x*y^3/3! ) + (-x^6/6! + 8*x^4*y^2/(4!*2!) - 8*x^3*y^3/(3!*3!) + x*y^5/5! ) + ( x^8/8! - 24*x^6*y^2/(6!*2!) + 24*x^3*y^5/(3!*5!) - x*y^7/7! ) + (-x^10/10! + 48*x^8*y^2/(8!*2!) - 576*x^6*y^4/(6!*4!) + 576*x^5*y^5/(5!*5!) - 48*x^3*y^7/(3!*7!) + x*y^9/9! ) + ( x^12/12! - 80*x^10*y^2/(10!*2!) + 3200*x^8*y^4/(8!*4!) - 3200*x^5*y^7/(5!*7!) + 80*x^3*y^9/(3!*9!) - x*y^11/11! ) + (-x^14/14! + 120*x^12*y^2/(12!*2!) - 10240*x^10*y^4/(10!*4!) + 160768*x^8*y^6/(8!*6!) - 160768*x^7*y^7/(7!*7!) + 10240*x^5*y^9/(5!*9!) - 120*x^3*y^11/(3!*11!) + x*y^13/13! ) + ...).
%e The e.g.f. of A326552 begins
%e SS(x*y) = (x*y) - 8*(x*y)^3/3!^2 + 576*(x*y)^5/5!^2 - 160768*(x*y)^7/7!^2 + 123535360*(x*y)^9/9!^2 - 212713734144*(x*y)^11/11!^2 + 716196297048064*(x*y)^13/13!^2 - 4280584942657732608*(x*y)^15/15!^2 + 42250703121584165486592*(x*y)^17/17!^2 - 651154631135458759089848320*(x*y)^19/19!^2 + 14983590319172065236171175755776*(x*y)^21/21!^2 + ... + A326552(n)*(x*y)^(2*n-1)/(2*n-1)! + ...
%e such that
%e SS(x*y) = Dx*Dy - Cx*Cy - Sx*Sy.
%e The e.g.f. of A326551 begins
%e CC(x*y) = 1 - 2*(x*y)^2/2!^2 + 56*(x*y)^4/4!^2 - 8336*(x*y)^6/6!^2 + 3985792*(x*y)^8/8!^2 - 4679517952*(x*y)^10/10!^2 + 11427218287616*(x*y)^12/12!^2 - 51793067942397952*(x*y)^14/14!^2 + 400951893341645930496*(x*y)^16/16!^2 - 4975999084909976839454720*(x*y)^18/18!^2 + 94178912073481319162642169856*(x*y)^20/20!^2 -+ ... + A326551(n)*(x*y)^(2*n)/(2*n)! + ...
%e such that
%e CC(x*y)^2 = (Cx*Dy + Cy*Dx)^2 + (Sx*Dy + Sy*Dx)^2 + (Sx*Cy - Sy*Cx)^2,
%e and CC(x*y)^2 + SS(x*y)^2 = 1.
%o (PARI)
%o {TSx(n,k) = my(Cx=1,Sx=x,Dx=1,Cy=1,Sy=y,Dy=1);
%o for(i=0,2*n+1,
%o Sx = intformal( Cx*Dy + Cy*Dx, x) + O(x^(2*n+2));
%o Cx = sqrt(1/2) - intformal( Sx*Dy + Sy*Dx, x);
%o Dx = sqrt(1/2) - intformal( Sx*Cy - Sy*Cx, x);
%o Sy = intformal( Cy*Dx + Cx*Dy, y) + O(y^(2*n+2));
%o Cy = sqrt(1/2) - intformal( Sy*Dx + Sx*Dy, y);
%o Dy = sqrt(1/2) - intformal( Sy*Cx - Sx*Cy, y);
%o );
%o round( (2*n-2*k+1)!*(2*k)! * polcoeff( polcoeff(Sx, 2*n-2*k+1,x),2*k,y) )}
%o for(n=0,10, for(k=0,n, print1( TSx(n,k),", "));print(""))
%Y Cf. A326801 (Cx), A326802 (Dx), A326803 (central terms).
%Y Cf. A326551 (CC), A326552 (SS), A326797.
%K sign,tabl
%O 1,5
%A _Paul D. Hanna_, Jul 27 2019
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