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%I #14 Feb 21 2024 10:10:14
%S 1,-1,0,1,12,0,-1,-420,-120,0,1,10248,36400,896,0,-1,-196920,-4858560,
%T -2170560,-5760,0,1,3247860,461126160,1127738304,102960000,33792,0,-1,
%U -48361404,-35248293080,-340884800256,-187282263168,-4083183104,-186368,0,1,669616080,2290777550880,76526954183680,153279541958400,25081621813248,141360128000,983040,0
%N Expansion of e.g.f. C(x,k) satisfying C(x,k) = cos( x*cos(k*x*C(x,k)) ), as a triangle read by rows.
%C The unsigned row sums equal A143601.
%C Signed version of triangle A370430.
%C A row reversal of triangle A370332.
%F E.g.f.: C(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n)*k^(2*j)/(2*n)! along with the related functions C = C(x,k), S = S(x,k), D = D(x,k), and T = T(x,k) satisfy the following formulas.
%F Definition.
%F (1.a) (C + i*S) = exp(i*x*D).
%F (1.b) (D + i*k*T) = exp(i*k*x*C).
%F (2.a) C^2 + S^2 = 1.
%F (2.b) D^2 + k^2*T^2 = 1.
%F Circular functions.
%F (3.a) C = cos(x*D).
%F (3.b) S = sin(x*D).
%F (3.c) D = cos(k*x*C).
%F (3.d) T = (1/k) * sin(k*x*C).
%F (4.a) C = cos( x*cos(k*x*C) ).
%F (4.b) S = sin( x*cos(k*x*sqrt(1 - S^2)) ).
%F (4.c) D = cos( k*x*cos(x*D) ).
%F (4.d) T = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T^2)) ).
%F (5.a) (C*D - k*S*T) = cos(x*D + k*x*C).
%F (5.b) (S*D + k*C*T) = sin(x*D + k*x*C).
%F Transformations.
%F (6.a) C(x, 1/k) = D(x/k, k).
%F (6.b) D(x, 1/k) = C(x/k, k).
%F (6.c) S(x, 1/k) = k * T(x/k, k).
%F (6.d) T(x, 1/k) = k * S(x/k, k).
%F (6.e) D(x, k) = C(k*x, 1/k).
%F (6.f) C(x, k) = D(k*x, 1/k).
%F (6.g) T(x, k) = (1/k) * S(k*x, 1/k).
%F (6.h) S(x, k) = (1/k) * T(k*x, 1/k).
%F Integrals.
%F (7.a) C = 1 - Integral S*D + x*S*D' dx.
%F (7.b) S = Integral C*D + x*C*D' dx.
%F (7.c) D = 1 - k^2 * Integral T*C + x*T*C' dx.
%F (7.d) T = Integral D*C + x*D*C' dx.
%F Derivatives (d/dx).
%F (8.a) C*C' = -S*S'.
%F (8.b) D*D' = -k^2*T*T'.
%F (9.a) C' = -S * (D + x*D').
%F (9.b) S' = C * (D + x*D').
%F (9.c) D' = -k^2 * T * (C + x*C').
%F (9.d) T' = D * (C + x*C').
%F (10.a) C' = -S * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
%F (10.b) S' = C * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
%F (10.c) D' = -k^2 * T * (C - x*S*D) / (1 - k^2*x^2*S*T).
%F (10.d) T' = D * (C - x*S*D) / (1 - k^2*x^2*S*T).
%F (11.a) (C + x*C') = (C - x*S*D) / (1 - k^2*x^2*S*T).
%F (11.b) (D + x*D') = (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
%e E.g.f.: C(x,k) = 1 - (1)*x^2/2! + (1 + 12*k^2)*x^4/4! - (1 + 420*k^2 + 120*k^4)*x^6/6! + (1 + 10248*k^2 + 36400*k^4 + 896*k^6)*x^8/8! - (1 + 196920*k^2 + 4858560*k^4 + 2170560*k^6 + 5760*k^8)*x^10/10! + (1 + 3247860*k^2 + 461126160*k^4 + 1127738304*k^6 + 102960000*k^8 + 33792*k^10)*x^12/12! - (1 + 48361404*k^2 + 35248293080*k^4 + 340884800256*k^6 + 187282263168*k^8 + 4083183104*k^10 + 186368*k^12)*x^14/14! + ...
%e where C(x,k) = cos( x*cos(k*x*C(x,k)) ).
%e This triangle of coefficients a(n,j) of x^(2*n)*k^(2*j)/(2*n)! in C(x,k) begins
%e 1;
%e -1, 0;
%e 1, 12, 0;
%e -1, -420, -120, 0;
%e 1, 10248, 36400, 896, 0;
%e -1, -196920, -4858560, -2170560, -5760, 0;
%e 1, 3247860, 461126160, 1127738304, 102960000, 33792, 0;
%e -1, -48361404, -35248293080, -340884800256, -187282263168, -4083183104, -186368, 0;
%e 1, 669616080, 2290777550880, 76526954183680, 153279541958400, 25081621813248, 141360128000, 983040, 0; ...
%o (PARI) {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n)));
%o for(i=1, 2*n,
%o C = cos( x*cos(k*x*C +Ox) );
%o S = sin( x*cos(k*x*sqrt(1 - S^2 +Ox)) );
%o D = cos( k*x*cos(x*D +Ox));
%o T = (1/k)*sin( k*x*cos(x*sqrt(1 - k^2*T^2 +Ox))););
%o (2*n)! * polcoeff(polcoeff(C, 2*n, x), 2*j, k)}
%o for(n=0, 10, for(j=0, n, print1( a(n, j), ", ")); print(""))
%Y Cf. A370331 (S), A370332 (D), A370333 (T).
%Y Cf. A370430.
%K sign,tabl
%O 0,5
%A _Paul D. Hanna_, Feb 19 2024