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%I #34 Sep 14 2024 12:30:28
%S 1,1,2,5,8,16,61,94,136,272,1385,2108,2840,3968,7936,50521,76474,
%T 100096,128704,176896,353792,2702765,4079408,5261120,6531968,8211200,
%U 11184128,22368256,199360981,300392854,384082456,468079984,566256256,702724864,951878656,1903757312,19391512145,29186948708,37114459400,44727839168,52998922880,63158486528,77747624960,104932671488,209865342976,2404879675441,3617015269234,4582440724816,5482201711264,6414037062016,7476185292544,8824137404416,10803399245824,14544442556416,29088885112832
%N E.g.f. S(y,x) = sin(y) / sqrt(1 - sin(x)^2 - sin(y)^2).
%C Row reversal of triangle A324610.
%C Related identity: cos(x+y)*cos(x-y) = (1 - sin(x)^2 - sin(y)^2). - _Paul D. Hanna_, Sep 14 2024
%C Name changed Sep 14 2024; prior name was: E.g.f. S(y,x) = Integral C(y,x)^2 * C(x,y) dy, where C(y,x)^2 - S(y,x)^2 = 1 and C(x,y) = Integral S(x,y)*C(x,y)*C(y,x) dx.
%H Paul D. Hanna, <a href="/A324612/b324612.txt">Table of n, a(n) for n = 0..495 terms in rows 0..30 of this triangle in flattened form.</a>
%F E.g.f. Sy = S(y,x) and related functions Cy = C(y,x), Sx = S(x,y), and Cx = C(x,y) satisfy the following relations.
%F (1a) Cx = 1 + Integral Sx * Cx*Cy dx.
%F (1b) Sx = Integral Cx * Cx*Cy dx.
%F (1c) Cy = 1 + Integral Sy * Cx*Cy dy.
%F (1d) Sy = Integral Cy * Cx*Cy dy.
%F (2a) Cx^2 - Sx^2 = 1.
%F (2b) Cy^2 - Sy^2 = 1.
%F (3a) Cx = cosh( Integral Cx*Cy dx ).
%F (3b) Sx = sinh( Integral Cx*Cy dx ).
%F (3c) Cy = cosh( Integral Cx*Cy dy ).
%F (3d) Sy = sinh( Integral Cx*Cy dy ).
%F (4a) Cx + Sx = exp( Integral Cx*Cy dx ).
%F (4b) Cy + Sy = exp( Integral Cx*Cy dy ).
%F (5a) (Cx + Sx)*(Cy + Sy) = (1 + sin(x+y))/cos(x+y).
%F (5b) (Cx + Sx)*(Cy - Sy) = (1 + sin(x-y))/cos(x-y).
%F (6a) Cx*Cy + Sx*Sy = 1/cos(x+y).
%F (6b) Cx*Sy + Sx*Cy = tan(x+y).
%F (7a) Cx*Cy = ( 1/cos(x+y) + 1/cos(x-y) )/2.
%F (7b) Sx*Sy = ( 1/cos(x+y) - 1/cos(x-y) )/2.
%F (7c) Cx*Sy = ( tan(x+y) - tan(x-y) )/2.
%F (7d) Sx*Cy = ( tan(x+y) + tan(x-y) )/2.
%F (8a) Cx*Cy = cos(x)*cos(y) / (cos(x+y)*cos(x-y)).
%F (8b) Sx*Sy = sin(x)*sin(y) / (cos(x+y)*cos(x-y)).
%F (8c) Cx*Sy = cos(y)*sin(y) / (cos(x+y)*cos(x-y)).
%F (8d) Sx*Cy = sin(x)*cos(x) / (cos(x+y)*cos(x-y)).
%F (9a) Cx + Sx = sqrt( (1 + sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ).
%F (9b) Cy + Sy = sqrt( (1 + sin(x+y))/cos(x+y) * (1 - sin(x-y))/cos(x-y) ).
%F (9c) Cx - Sx = sqrt( (1 - sin(x+y))/cos(x+y) * (1 - sin(x-y))/cos(x-y) ).
%F (9d) Cy - Sy = sqrt( (1 - sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ).
%F Let E(y,x) = sqrt( (1 + sin(x+y))/cos(x+y) * (1 - sin(x-y))/cos(x-y) ) then
%F (10a) E(y,x) = C(y,x) + S(y,x) where E(-y,x) = 1/E(y,x),
%F (10b) C(y,x) = (E(y,x) + E(-y,x))/2,
%F (10c) S(y,x) = (E(y,x) - E(-y,x))/2.
%F From _Paul D. Hanna_, Sep 14 2024: (Start) Explicitly,
%F (11a) Cx = cos(y) / sqrt(1 - sin(x)^2 - sin(y)^2).
%F (11b) Sx = sin(x) / sqrt(1 - sin(x)^2 - sin(y)^2).
%F (11c) Cy = cos(x) / sqrt(1 - sin(x)^2 - sin(y)^2).
%F (11d) Sy = sin(y) / sqrt(1 - sin(x)^2 - sin(y)^2).
%F (End)
%F PARTICULAR ARGUMENTS.
%F E.g.f. at x = 0: S(y,x=0) = tan(y).
%F E.g.f. at x = y: S(y,x=y) = sin(y)/sqrt(cos(2*y)).
%F FORMULAS INVOLVING TERMS.
%F T(n,0) = A000364(n) for n >= 0, where A000364 is the secant numbers.
%F T(n-1,n) = A000182(n) for n >= 1, where A000182 is the tangent numbers.
%e E.g.f.: S(y,x) = y + (1*x^2*y/2! + 2*y^3/3!) + (5*x^4*y/4! + 8*x^2*y^3/(2!*3!) + 16*y^5/5!) + (61*x^6*y/6! + 94*x^4*y^3/(4!*3!) + 136*x^2*y^5/(2!*5!) + 272*y^7/7!) + (1385*x^8*y/8! + 2108*x^6*y^3/(6!*3!) + 2840*x^4*y^5/(4!*5!) + 3968*x^2*y^7/(2!*7!) + 7936*y^9/9!) + (50521*x^10*y/10! + 76474*x^8*y^3/(8!*3!) + 100096*x^6*y^5/(6!*5!) + 128704*x^4*y^7/(4!*7!) + 176896*x^2*y^9/(2!*9!) + 353792*y^11/11!) + (2702765*x^12*y/12! + 4079408*x^10*y^3/(10!*3!) + 5261120*x^8*y^5/(8!*5!) + 6531968*x^6*y^7/(6!*7!) + 8211200*x^4*y^9/(4!*9!) + 11184128*x^2*y^11/(2!*11!) + 22368256*y^13/13!) + ...
%e such that S(y,x) = Integral C(y,x)^2 * C(x,y) dx.
%e Explicitly,
%e S(y,x) = ( sqrt( (1 + sin(x+y))/cos(x+y) * (1 - sin(x-y))/cos(x-y) ) - sqrt( (1 - sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ) )/2.
%e This triangle of coefficients T(n,k) of x^(2*n-2*k)*y^(2*k+1)/((2*n-2*k)!*(2*k+1)!) in e.g.f. S(y,x) begins
%e 1;
%e 1, 2;
%e 5, 8, 16;
%e 61, 94, 136, 272;
%e 1385, 2108, 2840, 3968, 7936;
%e 50521, 76474, 100096, 128704, 176896, 353792;
%e 2702765, 4079408, 5261120, 6531968, 8211200, 11184128, 22368256;
%e 199360981, 300392854, 384082456, 468079984, 566256256, 702724864, 951878656, 1903757312;
%e 19391512145, 29186948708, 37114459400, 44727839168, 52998922880, 63158486528, 77747624960, 104932671488, 209865342976; ...
%e RELATED SERIES.
%e S(x,y) = x + (2*x^3/3! + 1*x*y^2/2!) + (16*x^5/5! + 8*x^3*y^2/(3!*2!) + 5*x*y^4/4!) + (272*x^7/7! + 136*x^5*y^2/(5!*2!) + 94*x^3*y^4/(3!*4!) + 61*x*y^6/6!) + (7936*x^9/9! + 3968*x^7*y^2/(7!*2!) + 2840*x^5*y^4/(5!*4!) + 2108*x^3*y^6/(3!*6!) + 1385*x*y^8/8!) + (353792*x^11/11! + 176896*x^9*y^2/(9!*2!) + 128704*x^7*y^4/(7!*4!) + 100096*x^5*y^6/(5!*6!) + 76474*x^3*y^8/(3!*8!) + 50521*x*y^10/10!) + (22368256*x^13/13! + 11184128*x^11*y^2/(11!*2!) + 8211200*x^9*y^4/(9!*4!) + 6531968*x^7*y^6/(7!*6!) + 5261120*x^5*y^8/(5!*8!) + 4079408*x^3*y^10/(3!*10!) + 2702765*x*y^12/12!) + ...
%e such that C(x,y) = cosh( Integral C(x,y)*C(y,x) dx ).
%e C(y,x) = 1 + (1*y^2/2!) + (2*x^2*y^2/(2!*2!) + 5*y^4/4!) + (16*x^4*y^2/(4!*2!) + 28*x^2*y^4/(2!*4!) + 61*y^6/6!) + (272*x^6*y^2/(6!*2!) + 440*x^4*y^4/(4!*4!) + 662*x^2*y^6/(2!*6!) + 1385*y^8/8!) + (7936*x^8*y^2/(8!*2!) + 12448*x^6*y^4/(6!*4!) + 17176*x^4*y^6/(4!*6!) + 24568*x^2*y^8/(2!*8!) + 50521*y^10/10!) + (353792*x^10*y^2/(10!*2!) + 546560*x^8*y^4/(8!*4!) + 727232*x^6*y^6/(6!*6!) + 949520*x^4*y^8/(4!*8!) + 1326122*x^2*y^10/(2!*10!) + 2702765*y^12/12!) + ...
%e such that C(y,x)^2 - S(y,x)^2 = 1.
%e C(x,y) = 1 + (1*x^2/2!) + (5*x^4/4! + 2*x^2*y^2/(2!*2!)) + (61*x^6/6! + 28*x^4*y^2/(4!*2!) + 16*x^2*y^4/(2!*4!)) + (1385*x^8/8! + 662*x^6*y^2/(6!*2!) + 440*x^4*y^4/(4!*4!) + 272*x^2*y^6/(2!*6!)) + (50521*x^10/10! + 24568*x^8*y^2/(8!*2!) + 17176*x^6*y^4/(6!*4!) + 12448*x^4*y^6/(4!*6!) + 7936*x^2*y^8/(2!*8!)) + (2702765*x^12/12! + 1326122*x^10*y^2/(10!*2!) + 949520*x^8*y^4/(8!*4!) + 727232*x^6*y^6/(6!*6!) + 546560*x^4*y^8/(4!*8!) + 353792*x^2*y^10/(2!*10!)) + ...
%e such that C(x,y) = cosh( Integral C(x,y)*C(y,x) dx ).
%o (PARI) {T(n,k) = my(Cx = 1 + x*O(x^(2*n)), Cy = 1 + y*O(y^(2*n)));
%o for(i=1,2*n,
%o Cx = cosh(intformal(Cx*Cy,x));
%o Cy = cosh(intformal(Cx*Cy,y)););
%o Sx = sinh(intformal(Cx*Cy,x));
%o Sy = sinh(intformal(Cx*Cy,y));
%o (2*n-2*k)!*(2*k+1)! * polcoeff(polcoeff(Sy,2*n-2*k,x),2*k+1,y)}
%o for(n=0,10,for(k=0,n, print1( T(n,k),", "));print(""))
%Y Cf. A324611 (C(y,x)), A324610 (S(x,y)), A324609 (C(x,y)).
%Y Cf. A000364 (T(n,0)), A000182 (T(n-1,n)).
%K nonn,tabl
%O 0,3
%A _Paul D. Hanna_, Mar 09 2019