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%I #20 Mar 06 2019 12:53:17
%S 1,1,3,1,50,5,1,693,1155,7,1,9972,70686,23268,9,1,135575,3479850,
%T 4871790,406725,11,1,1727622,157346475,631853508,283223655,6334614,13,
%U 1,20926185,6802724565,67722059405,87071219235,14965994043,90680135,15,1,243932456,282646403340,6596182917688,19436510145910,10365430299224,734880648684,1219662280,17,1,2760372459,11263126697316,606536559381564,3683652871295358,4502242398249882,1126425038851476,33789380091948,15642110601,19
%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)!, as a triangle of coefficients T(n,k) read by rows.
%C See A322220 for another description of the e.g.f. of this sequence:
%C T(n,k) = binomial(2*n+1,2*k) * A322220(n,k).
%H Paul D. Hanna, <a href="/A322730/b322730.txt">Table of n, a(n) for n = 0..1274 terms of this triangle for 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)! begins
%e S(x,y) = x + (1*x^3 + 3*x*y^2)/3! + (1*x^5 + 50*x^3*y^2 + 5*x*y^4)/5! + (1*x^7 + 693*x^5*y^2 + 1155*x^3*y^4 + 7*x*y^6)/7! + (1*x^9 + 9972*x^7*y^2 + 70686*x^5*y^4 + 23268*x^3*y^6 + 9*x*y^8)/9! + (1*x^11 + 135575*x^9*y^2 + 3479850*x^7*y^4 + 4871790*x^5*y^6 + 406725*x^3*y^8 + 11*x*y^10)/11! + (1*x^13 + 1727622*x^11*y^2 + 157346475*x^9*y^4 + 631853508*x^7*y^6 + 283223655*x^5*y^8 + 6334614*x^3*y^10 + 13*x*y^12)/13! + (1*x^15 + 20926185*x^13*y^2 + 6802724565*x^11*y^4 + 67722059405*x^9*y^6 + 87071219235*x^7*y^8 + 14965994043*x^5*y^10 + 90680135*x^3*y^12 + 15*x*y^14)/15! + ...
%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)! in S(x,y) starts
%e 1;
%e 1, 3;
%e 1, 50, 5;
%e 1, 693, 1155, 7;
%e 1, 9972, 70686, 23268, 9;
%e 1, 135575, 3479850, 4871790, 406725, 11;
%e 1, 1727622, 157346475, 631853508, 283223655, 6334614, 13;
%e 1, 20926185, 6802724565, 67722059405, 87071219235, 14965994043, 90680135, 15;
%e 1, 243932456, 282646403340, 6596182917688, 19436510145910, 10365430299224, 734880648684, 1219662280, 17;
%e 1, 2760372459, 11263126697316, 606536559381564, 3683652871295358, 4502242398249882, 1126425038851476, 33789380091948, 15642110601, 19; ...
%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)! *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. A322731, A322732, A322733 (row sums).
%Y Cf. A322220.
%K nonn,tabl
%O 0,3
%A _Paul D. Hanna_, Dec 26 2018
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