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A370333
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Expansion of e.g.f. T(x,k) satisfying T(x,k) = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T(x,k)^2)) ), as a triangle read by rows.
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4
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1, -3, -1, 5, 90, 1, -7, -3675, -2205, -1, 9, 107604, 532350, 46116, 1, -11, -2436885, -74042430, -52887450, -812295, -1, 13, 46444398, 7663602375, 24609789204, 4257556875, 12666654, 1, -15, -785872815, -643910782515, -7510986678195, -5841878527485, -292686719325, -181355265, -1
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OFFSET
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0,2
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COMMENTS
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Signed version of triangle A370433.
A row reversal of triangle A370331.
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LINKS
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FORMULA
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E.g.f.: T(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.
Definition.
(1.a) (C + i*S) = exp(i*x*D).
(1.b) (D + i*k*T) = exp(i*k*x*C).
(2.a) C^2 + S^2 = 1.
(2.b) D^2 + k^2*T^2 = 1.
Circular functions.
(3.a) C = cos(x*D).
(3.b) S = sin(x*D).
(3.c) D = cos(k*x*C).
(3.d) T = (1/k) * sin(k*x*C).
(4.a) C = cos( x*cos(k*x*C) ).
(4.b) S = sin( x*cos(k*x*sqrt(1 - S^2)) ).
(4.c) D = cos( k*x*cos(x*D) ).
(4.d) T = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T^2)) ).
(5.a) (C*D - k*S*T) = cos(x*D + k*x*C).
(5.b) (S*D + k*C*T) = sin(x*D + k*x*C).
Transformations.
(6.a) C(x, 1/k) = D(x/k, k).
(6.b) D(x, 1/k) = C(x/k, k).
(6.c) S(x, 1/k) = k * T(x/k, k).
(6.d) T(x, 1/k) = k * S(x/k, k).
(6.e) D(x, k) = C(k*x, 1/k).
(6.f) C(x, k) = D(k*x, 1/k).
(6.g) T(x, k) = (1/k) * S(k*x, 1/k).
(6.h) S(x, k) = (1/k) * T(k*x, 1/k).
Integrals.
(7.a) C = 1 - Integral S*D + x*S*D' dx.
(7.b) S = Integral C*D + x*C*D' dx.
(7.c) D = 1 - k^2 * Integral T*C + x*T*C' dx.
(7.d) T = Integral D*C + x*D*C' dx.
Derivatives (d/dx).
(8.a) C*C' = -S*S'.
(8.b) D*D' = -k^2*T*T'.
(9.a) C' = -S * (D + x*D').
(9.b) S' = C * (D + x*D').
(9.c) D' = -k^2 * T * (C + x*C').
(9.d) T' = D * (C + x*C').
(10.a) C' = -S * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.b) S' = C * (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
(10.c) D' = -k^2 * T * (C - x*S*D) / (1 - k^2*x^2*S*T).
(10.d) T' = D * (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.a) (C + x*C') = (C - x*S*D) / (1 - k^2*x^2*S*T).
(11.b) (D + x*D') = (D - k^2*x*T*C) / (1 - k^2*x^2*S*T).
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EXAMPLE
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E.g.f.: T(x,k) = x - (3 + k^2)*x^3/3! + (5 + 90*k^2 + k^4)*x^5/5! - (7 + 3675*k^2 + 2205*k^4 + k^6)*x^7/7! + (9 + 107604*k^2 + 532350*k^4 + 46116*k^6 + k^8)*x^9/9! - (11 + 2436885*k^2 + 74042430*k^4 + 52887450*k^6 + 812295*k^8 + k^10)*x^11/11! + (13 + 46444398*k^2 + 7663602375*k^4 + 24609789204*k^6 + 4257556875*k^8 + 12666654*k^10 + k^12)*x^13/13! + ...
where T(x,k) = (1/k) * sin( k*x*cos(x*sqrt(1 - k^2*T(x,k)^2)) ).
This triangle of coefficients a(n,j) of x^(2*n+1)*k^(2*j)/(2*n+1)! in T(x,k) begins
1;
-3, -1;
5, 90, 1;
-7, -3675, -2205, -1;
9, 107604, 532350, 46116, 1;
-11, -2436885, -74042430, -52887450, -812295, -1;
13, 46444398, 7663602375, 24609789204, 4257556875, 12666654, 1;
-15, -785872815, -643910782515, -7510986678195, -5841878527485, -292686719325, -181355265, -1;
17, 12196578600, 45911000082220, 1766457334617976, 4451226370197750, 1124109212938712, 17658076954700, 2439315720, 1; ...
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PROG
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(PARI) {a(n, j) = my(C=1, S=x, D=1, T=x, Ox=x*O(x^(2*n+1)));
for(i=1, 2*n+1,
C = cos( x*cos(k*x*C +Ox) );
S = sin( x*cos(k*x*sqrt(1 - S^2 +Ox)) );
D = cos( k*x*cos(x*D +Ox));
T = (1/k)*sin( k*x*cos(x*sqrt(1 - k^2*T^2 +Ox))); );
(2*n+1)! *polcoeff(polcoeff(T, 2*n+1, x), 2*j, k)}
for(n=0, 10, for(k=0, n, print1( a(n, k), ", ")); print(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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