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A370331 Expansion of e.g.f. S(x,k) satisfying S(x,k) = sin( x*cos(k*x*sqrt(1 - S(x,k)^2)) ), as a triangle read by rows.

%I A370331 #11 Feb 21 2024 08:25:19
%S A370331 1,-1,-3,1,90,5,-1,-2205,-3675,-7,1,46116,532350,107604,9,-1,-812295,
%T A370331 -52887450,-74042430,-2436885,-11,1,12666654,4257556875,24609789204,
%U A370331 7663602375,46444398,13,-1,-181355265,-292686719325,-5841878527485,-7510986678195,-643910782515,-785872815,-15
%N A370331 Expansion of e.g.f. S(x,k) satisfying S(x,k) = sin( x*cos(k*x*sqrt(1 - S(x,k)^2)) ), as a triangle read by rows.
%C A370331 The unsigned row sums equal A007106.
%C A370331 Signed version of triangle A370431.
%C A370331 A row reversal of triangle A370333.
%F A370331 E.g.f.: S(x,k) = Sum_{n>=0} Sum_{j=0..n} a(n,j) * x^(2*n+1)*k^(2*j)/(2*n+1)! 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 A370331 Definition.
%F A370331 (1.a) (C + i*S) = exp(i*x*D).
%F A370331 (1.b) (D + i*k*T) = exp(i*k*x*C).
%F A370331 (2.a) C^2 + S^2 = 1.
%F A370331 (2.b) D^2 + k^2*T^2 = 1.
%F A370331 Circular functions.
%F A370331 (3.a) C = cos(x*D).
%F A370331 (3.b) S = sin(x*D).
%F A370331 (3.c) D = cos(k*x*C).
%F A370331 (3.d) T = (1/k) * sin(k*x*C).
%F A370331 (4.a) C = cos( x*cos(k*x*C) ).
%F A370331 (4.b) S = sin( x*cos(k*x*sqrt(1 - S^2)) ).
%F A370331 (4.c) D = cos( k*x*cos(x*D) ).
%F A370331 (4.d) T = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T^2)) ).
%F A370331 (5.a) (C*D - k*S*T) = cos(x*D + k*x*C).
%F A370331 (5.b) (S*D + k*C*T) = sin(x*D + k*x*C).
%F A370331 Transformations.
%F A370331 (6.a) C(x, 1/k) = D(x/k, k).
%F A370331 (6.b) D(x, 1/k) = C(x/k, k).
%F A370331 (6.c) S(x, 1/k) = k * T(x/k, k).
%F A370331 (6.d) T(x, 1/k) = k * S(x/k, k).
%F A370331 (6.e) D(x, k) = C(k*x, 1/k).
%F A370331 (6.f) C(x, k) = D(k*x, 1/k).
%F A370331 (6.g) T(x, k) = (1/k) * S(k*x, 1/k).
%F A370331 (6.h) S(x, k) = (1/k) * T(k*x, 1/k).
%F A370331 Integrals.
%F A370331 (7.a) C = 1 - Integral S*D + x*S*D' dx.
%F A370331 (7.b) S = Integral C*D + x*C*D' dx.
%F A370331 (7.c) D = 1 - k^2 * Integral T*C + x*T*C' dx.
%F A370331 (7.d) T = Integral D*C + x*D*C' dx.
%F A370331 Derivatives (d/dx).
%F A370331 (8.a) C*C' = -S*S'.
%F A370331 (8.b) D*D' = -k^2*T*T'.
%F A370331 (9.a) C' = -S * (D + x*D').
%F A370331 (9.b) S' = C * (D + x*D').
%F A370331 (9.c) D' = -k^2 * T * (C + x*C').
%F A370331 (9.d) T' = D * (C + x*C').
%F A370331 (10.a) C' = -S * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
%F A370331 (10.b) S' = C * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
%F A370331 (10.c) D' = -k^2 * T * (C - x*S*D) / (1 - k^2*x^2*S*T).
%F A370331 (10.d) T' = D * (C - x*S*D) / (1 - k^2*x^2*S*T).
%F A370331 (11.a) (C + x*C') = (C - x*S*D) / (1 - k^2*x^2*S*T).
%F A370331 (11.b) (D + x*D') = (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
%e A370331 E.g.f.: S(x,k) = x - (1 + 3*k^2)*x^3/3! + (1 + 90*k^2 + 5*k^4)*x^5/5! - (1 + 2205*k^2 + 3675*k^4 + 7*k^6)*x^7/7! + (1 + 46116*k^2 + 532350*k^4 + 107604*k^6 + 9*k^8)*x^9/9! - (1 + 812295*k^2 + 52887450*k^4 + 74042430*k^6 + 2436885*k^8 + 11*k^10)*x^11/11! + (1 + 12666654*k^2 + 4257556875*k^4 + 24609789204*k^6 + 7663602375*k^8 + 46444398*k^10 + 13*k^12)*x^13/13! + ...
%e A370331 where S(x,k) = sin( x*cos(k*x*sqrt(1 - S(x,k)^2)) ).
%e A370331 This triangle of coefficients a(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in S(x,k) begins
%e A370331  1;
%e A370331  -1, -3;
%e A370331  1, 90, 5;
%e A370331  -1, -2205, -3675, -7;
%e A370331  1, 46116, 532350, 107604, 9;
%e A370331  -1, -812295, -52887450, -74042430, -2436885, -11;
%e A370331  1, 12666654, 4257556875, 24609789204, 7663602375, 46444398, 13;
%e A370331  -1, -181355265, -292686719325, -5841878527485, -7510986678195, -643910782515, -785872815, -15;
%e A370331  1, 2439315720, 17658076954700, 1124109212938712, 4451226370197750, 1766457334617976, 45911000082220, 12196578600, 17; ...
%o A370331 (PARI) {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n+1)));
%o A370331 for(i=1, 2*n+1,
%o A370331 C = cos( x*cos(k*x*C +Ox) );
%o A370331 S = sin( x*cos(k*x*sqrt(1 - S^2 +Ox)) );
%o A370331 D = cos( k*x*cos(x*D +Ox));
%o A370331 T = (1/k)*sin( k*x*cos(x*sqrt(1 - k^2*T^2 +Ox))););
%o A370331 (2*n+1)! *polcoeff(polcoeff(S, 2*n+1, x), 2*j, k)}
%o A370331 for(n=0, 10, for(k=0, n, print1( a(n, k), ", ")); print(""))
%Y A370331 Cf. A370330 (C), A370332 (D), A370333 (T).
%Y A370331 Cf. A370431.
%K A370331 sign,tabl
%O A370331 0,3
%A A370331 _Paul D. Hanna_, Feb 19 2024