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EXAMPLE
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E.g.f.: A(x) = x - 3*x^3/(3!*2!) + 145*x^5/(5!*4!) - 30387*x^7/(7!*6!) + 18679617*x^9/(9!*8!) - 26803260803*x^11/(11!*10!) + 77353020714385*x^13/(13!*12!) -+ ...
The triangle of coefficients A326800(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
1;
-1, -1;
1, -3, 1;
-1, 15, 15, -1;
1, -35, 145, -35, 1;
-1, 63, -1505, -1505, 63, -1;
1, -99, 5985, -30387, 5985, -99, 1;
-1, 143, -16401, 539679, 539679, -16401, 143, -1;
1, -195, 36465, -3275811, 18679617, -3275811, 36465, -195, 1;
-1, 255, -70785, 12723711, -506849409, -506849409, 12723711, -70785, 255, -1;
1, -323, 124865, -38067315, 4363117473, -26803260803, 4363117473, -38067315, 124865, -323, 1; ...
in which the central terms form this sequence.
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! ) + ...
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PROG
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(PARI)
{a(n) = my(Cx=1, Sx=x, Dx=1, Cy=1, Sy=y, Dy=1);
for(i=0, 4*n+2,
Sx = intformal( Cx*Dy + Cy*Dx, x) + O(x^(2*n+2));
Cx = sqrt(1/2) - intformal( Sx*Dy + Sy*Dx, x);
Dx = sqrt(1/2) - intformal( Sx*Cy - Sy*Cx, x);
Sy = intformal( Cy*Dx + Cx*Dy, y) + O(y^(2*n+2));
Cy = sqrt(1/2) - intformal( Sy*Dx + Sx*Dy, y);
Dy = sqrt(1/2) - intformal( Sy*Cx - Sx*Cy, y);
);
round( (2*n+1)!*(2*n)! * polcoeff( polcoeff(Sx, 2*n+1, x), 2*n, y) )}
for(n=0, 15, print1( a(n), ", "))
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