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%I #62 Nov 21 2023 09:36:56
%S 1,1,1,1,5,1,1,33,33,1,1,277,561,277,1,1,2465,10545,10545,2465,1,1,
%T 22149,220065,368213,220065,22149,1,1,199297,4983681,13530881,
%U 13530881,4983681,199297,1,1,1793621,118758993,532981813,799527361,532981813,118758993,1793621,1,1,16142529,2905863441,22355025777,48737171169,48737171169,22355025777,2905863441,16142529,1,1,145282693,71982945345,986147170485,3116925490785,4360363161285,3116925490785,986147170485,71982945345,145282693,1
%N 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, where 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)!), as a triangle of coefficients T(n,k) read by rows.
%C See A322730 for another description of the e.g.f. of this sequence:
%C T(n,k) = A322730(n,k)/binomial(2*n+1,2*k).
%C This triangle is an unsigned version of A367380.
%H Paul D. Hanna, <a href="/A322220/b322220.txt">Table of n, a(n) for n = 0..1274 terms of this triangle as read by rows 0..50</a>
%F The special functions S(x,y), C(x,y), and D(x,y) satisfy the following relations.
%F (1a) S(x,y) = Integral C(x,y) * C(y,x) dx.
%F (1b) S(y,x) = Integral C(y,x) * C(x,y) dy.
%F (1c) C(x,y) = 1 + Integral S(x,y) * C(y,x) dx.
%F (1d) C(y,x) = 1 + Integral S(y,x) * C(x,y) dy.
%F (2a) C(x,y)^2 - S(x,y)^2 = 1.
%F (2b) C(y,x)^2 - S(y,x)^2 = 1.
%F (3a) S(x,y) = sinh( Integral C(y,x) dx ).
%F (3b) S(y,x) = sinh( Integral C(x,y) dy ).
%F (3c) C(x,y) = cosh( Integral C(y,x) dx ).
%F (3d) C(y,x) = cosh( Integral C(x,y) dy ).
%F (4a) C(x,y) + S(x,y) = exp( Integral C(y,x) dx ).
%F (4b) C(y,x) + S(y,x) = exp( Integral C(x,y) dy ).
%F (5a) d/dx S(x,y) = C(x,y) * C(y,x).
%F (5b) d/dx C(x,y) = S(x,y) * C(y,x).
%F (5c) d/dy S(y,x) = C(y,x) * C(x,y).
%F (5d) d/dy C(y,x) = S(y,x) * C(x,y).
%F Introducing function D(x,y) completes the symmetric relations as follows.
%F (6a) D(x,y) = Integral S(y,x) * C(x,y) dx.
%F (6b) D(y,x) = Integral S(x,y) * C(y,x) dy.
%F (7a) S(x,y) = sinh(x) + Integral C(x,y) * D(x,y) dy.
%F (7b) S(y,x) = sinh(y) + Integral C(y,x) * D(y,x) dx.
%F (7c) C(x,y) = cosh(x) + Integral S(x,y) * D(x,y) dy.
%F (7d) C(y,x) = cosh(y) + Integral S(y,x) * D(y,x) dx.
%F (8a) C(x,y) + S(x,y) = exp( x + Integral D(x,y) dy ).
%F (8b) C(y,x) + S(y,x) = exp( y + Integral D(y,x) dx ).
%F (9a) Integral C(y,x) dx = x + Integral D(x,y) dy.
%F (9b) Integral C(x,y) dy = y + Integral D(y,x) dx.
%F (10a) d/dy S(x,y) = C(x,y) * D(x,y).
%F (10b) d/dy C(x,y) = S(x,y) * D(x,y).
%F (10c) d/dx S(y,x) = C(y,x) * D(y,x).
%F (10d) d/dx C(y,x) = S(y,x) * D(y,x).
%F (10e) d/dx D(x,y) = S(y,x) * C(x,y).
%F (10f) d/dy D(y,x) = S(x,y) * C(y,x).
%F For brevity, let Cx = C(x,y), Cy = C(y,x), Sx = S(x,y), Sy = S(y,x), Dx = D(x,y), Dy = D(y,x), then further relations may be written as follows.
%F (11a) Cx*Cy + Sx*Sy = cosh(y) + Integral (Cy + Dy)*(Sx*Cy + Cx*Sy) dx.
%F (11b) Sx*Cy + Cx*Sy = sinh(y) + Integral (Cy + Dy)*(Cx*Cy + Sx*Sy) dx.
%F (11c) Cx*Cy + Sx*Sy = cosh(x) + Integral (Cx + Dx)*(Sx*Cy + Cx*Sy) dy.
%F (11d) Sx*Cy + Cx*Sy = sinh(x) + Integral (Cx + Dx)*(Cx*Cy + Sx*Sy) dy.
%F (12a) (Cx + Sx)*(Cy + Sy) = exp( y + Integral Cy + Dy dx ).
%F (12b) (Cx + Sx)*(Cy + Sy) = exp( x + Integral Cx + Dx dy ).
%F (12c) (Cx + Sx)*(Cy + Sy) = exp( x + y + Integral Dx dy + Integral Dy dx ).
%F (12d) (Cx + Sx)*(Cy + Sy) = exp( x + y + Integral Integral Sx*Cy + Cx*Sy dx dy ).
%F (12e) x + Integral (Cx + Dx) dy = y + Integral (Cy + Dy) dx.
%F (13a) d/dx (Cx + Sx)*(Cy + Sy) = (Cx + Sx)*(Cy + Sy)*(Cy + Dy).
%F (13b) d/dy (Cx + Sx)*(Cy + Sy) = (Cx + Sx)*(Cy + Sy)*(Cx + Dx).
%F (14a) (Cx + Sx)*(Cy + Sy) = exp(y) + Integral (Cx + Sx)*(Cy + Sy)*(Cy + Dy) dx.
%F (14b) (Cx + Sx)*(Cy + Sy) = exp(x) + Integral (Cx + Sx)*(Cy + Sy)*(Cx + Dx) dy.
%e 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
%e S(x,y) = x + (1*x^3/3! + 1*x*y^2/2!) + (1*x^5/5! + 5*x^3*y^2/(3!*2!) + 1*x*y^4/4!) + (1*x^7/7! + 33*x^5*y^2/(5!*2!) + 33*x^3*y^4/(3!*4!) + 1*x*y^6/6!) + (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^8/8!) + (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^10/10!) + (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^12/12!) + (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^14/14!) + ...
%e The series S(x,y) may be defined by
%e S(x,y) = Integral C(x,y) * C(y,x) dx, and
%e S(y,x) = Integral C(y,x) * C(x,y) dy,
%e such that C(x,y)^2 = 1 + S(x,y)^2.
%e TRIANGLE.
%e 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) starts
%e 1;
%e 1, 1;
%e 1, 5, 1;
%e 1, 33, 33, 1;
%e 1, 277, 561, 277, 1;
%e 1, 2465, 10545, 10545, 2465, 1;
%e 1, 22149, 220065, 368213, 220065, 22149, 1;
%e 1, 199297, 4983681, 13530881, 13530881, 4983681, 199297, 1;
%e 1, 1793621, 118758993, 532981813, 799527361, 532981813, 118758993, 1793621, 1;
%e 1, 16142529, 2905863441, 22355025777, 48737171169, 48737171169, 22355025777, 2905863441, 16142529, 1;
%e 1, 145282693, 71982945345, 986147170485, 3116925490785, 4360363161285, 3116925490785, 986147170485, 71982945345, 145282693, 1; ...
%e RELATED SERIES.
%e The related series C(x,y), where C(x,y)^2 - S(x,y)^2 = 1, begins
%e 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!)) + ...
%e This series may be defined by
%e C(x,y) = cosh( Integral C(y,x) dx ), and
%e C(y,x) = cosh( Integral C(x,y) dy ).
%e The related series D(x,y) = Integral S(y,x) * C(x,y) dx, begins
%e D(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!) + ...
%o (PARI) {T(n,k) = my(Sx=x,Sy=y,Cx=1,Cy=1); for(i=1,2*n,
%o Sx = intformal( Cx*Cy +x*O(x^(2*n)), x);
%o Cx = 1 + intformal( Sx*Cy, x);
%o Sy = intformal( Cy*Cx +y*O(y^(2*k)), y);
%o Cy = 1 + intformal( Sy*Cx, y));
%o (2*n+1-2*k)!*(2*k)! *polcoeff(polcoeff(Sx,2*n+1-2*k,x),2*k,y)}
%o for(n=0,10,for(k=0,n,print1( T(n,k),", "));print(""))
%Y Cf. A322221, A322222, A322223, A322224, A325153 (column 1).
%Y Cf. A322730, A367380.
%Y Cf. A324610 (variant).
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Dec 23 2018
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