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%I #12 Feb 20 2020 13:57:01
%S 1,0,1,0,12,1,0,120,180,1,0,896,9520,2632,1,0,5760,369600,504000,
%T 37080,1,0,33792,12735360,57376704,23562000,487476,1,0,186368,
%U 399246848,5256451200,6645999360,1039654616,6045676,1,0,983040,11498905600,427859367936,1336544352000,674286412800,44074736160,71745360,1,0,5013504,308405698560,32219490680832,224718683904000,271641150443520,63624274826112,1793634471360,823269744,1
%N E.g.f. C(y,x) = 1 + Integral S(y,x)*C(x,y) dy such that C(x,y)^2 - S(x,y)^2 = 1 and C(x,y) = Integral S(x,y)*C(y,x) dx, where C(y,x) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k)*y^(2*k)/(2*n)!, as a triangle of coefficients T(n,k) read by rows.
%C See A322222 for another description of the e.g.f. of this sequence:
%C T(n,k) = binomial(2*n,2*k) * A322222(n,k).
%C Row reversal of A322731.
%H Paul D. Hanna, <a href="/A322732/b322732.txt">Table of n, a(n) for n = 0..1274 terms of this triangle 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. C(y,x) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k)*y^(2*k)/(2*n)! begins
%e C(y,x) = 1 + (1*y^2)/2! + (12*x^2*y^2 + 1*y^4)/4! + (120*x^4*y^2 + 180*x^2*y^4 + 1*y^6)/6! + (896*x^6*y^2 + 9520*x^4*y^4 + 2632*x^2*y^6 + 1*y^8)/8! + (5760*x^8*y^2 + 369600*x^6*y^4 + 504000*x^4*y^6 + 37080*x^2*y^8 + 1*y^10)/10! + (33792*x^10*y^2 + 12735360*x^8*y^4 + 57376704*x^6*y^6 + 23562000*x^4*y^8 + 487476*x^2*y^10 + 1*y^12)/12! + (186368*x^12*y^2 + 399246848*x^10*y^4 + 5256451200*x^8*y^6 + 6645999360*x^6*y^8 + 1039654616*x^4*y^10 + 6045676*x^2*y^12 + 1*y^14)/14! + (983040*x^14*y^2 + 11498905600*x^12*y^4 + 427859367936*x^10*y^6 + 1336544352000*x^8*y^8 + 674286412800*x^6*y^10 + 44074736160*x^4*y^12 + 71745360*x^2*y^14 + 1*y^16)/16! + ...
%e This series may be defined by
%e C(y,x) = cosh( Integral C(x,y) dy ), and
%e C(x,y) = cosh( Integral C(y,x) dx ).
%e TRIANGLE.
%e This triangle of coefficients T(n,k) of x^(2*n-2*k)*y^(2*k)/(2*n)! in C(y,x) starts
%e 1;
%e 0, 1;
%e 0, 12, 1;
%e 0, 120, 180, 1;
%e 0, 896, 9520, 2632, 1;
%e 0, 5760, 369600, 504000, 37080, 1;
%e 0, 33792, 12735360, 57376704, 23562000, 487476, 1;
%e 0, 186368, 399246848, 5256451200, 6645999360, 1039654616, 6045676, 1;
%e 0, 983040, 11498905600, 427859367936, 1336544352000, 674286412800, 44074736160, 71745360, 1;
%e 0, 5013504, 308405698560, 32219490680832, 224718683904000, 271641150443520, 63624274826112, 1793634471360, 823269744, 1; ...
%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)! *polcoeff(polcoeff(Cy, 2*n-2*k, x), 2*k, y)}
%o for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))
%Y Cf. A322730, A322731.
%Y Cf. A322222.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Jan 01 2019
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