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A326798 Consider the e.g.f. B(x,y) = Sum_{n>=0} Sum_{k=0..floor(n/2)} T(n,k) * x^(2*n-2*k) * y^(2*k) / (2*n)! and related functions A(x,y) and C(x,y), as defined in the Formula section. Sequence gives the triangular array of coefficients T(n,k) (n>=0, 0<=k<=floor(n/2)) of B(x,y). 6
1, -1, 1, 0, -1, 120, 1, -672, 0, -1, 2160, -120960, 1, -5280, 1584000, 0, -1, 10920, -10250240, 482786304, 1, -20160, 45427200, -11480268800, 0, -1, 34272, -157651200, 124816613376, -5405660282880, 1, -54720, 460158720, -875447623680, 203526629130240, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
The e.g.f. B(x,y) at y = x is described by A326795.
LINKS
FORMULA
The e.g.f. Bx = B(x,y) = Sum_{n>=0} Sum_{k=0..floor(n/2)} T(n,k)*x^(2*n-2*k)*y^(2*k)/(2*n)! and related functions Ax = A(x,y), Cx = C(x,y), Ay = A(y,x), By = B(y,x), and Cy = C(y,x) satisfy the following relations.
DEFINITION.
(1a) Ax = 0 + Integral Bx*Cy - Cx*By dx,
(1b) Bx = 1 + Integral Cx*Ay - Ax*Cy dx,
(1c) Cx = 0 + Integral Ax*By - Bx*Ay dx.
(2a) Ay = 0 + Integral By*Cx - Cy*Bx dy,
(2b) By = 0 + Integral Cy*Ax - Ay*Cx dy,
(2c) Cy = 1 + Integral Ay*Bx - By*Ax dy.
IDENTITIES.
(3a) Ax^2 + Bx^2 + Cx^2 = 1.
(3b) Ay^2 + By^2 + Cy^2 = 1.
(4a) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2 = 1.
(4b) (Ax*Ay + Bx*By + Cx*Cy)^2 + (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2 = 1.
(5a) Ax*(d/dx Ax) + Bx*(d/dx Bx) + Cx*(d/dx Cx) = 0.
(5b) Ay*(d/dy Ay) + By*(d/dy By) + Cy*(d/dy Cy) = 0.
(5c) Ax*(d/dy Ay) + Bx*(d/dy By) + Cx*(d/dy Cy) = 0.
(5d) Ay*(d/dx Ax) + By*(d/dx Bx) + Cy*(d/dx Cx) = 0.
(5e) Ax*(d/dy Ax) + Bx*(d/dy Bx) + Cx*(d/dy Cx) = 0.
(5f) Ay*(d/dx Ay) + By*(d/dx By) + Cy*(d/dx Cy) = 0.
RELATED FUNCTIONS.
(6a) SS(x*y) = Ax*Ay + Bx*By + Cx*Cy.
(6b) d/dx SS(x*y) = Ax*(d/dx Ay) + Bx*(d/dx By) + Cx*(d/dx Cy).
(6c) d/dy SS(x*y) = Ay*(d/dy Ax) + By*(d/dy Bx) + Cy*(d/dy Cx).
(7a) CC(x*y)^2 = (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2.
(7b) CC(x*y)^2 = (d/dx Ax)^2 + (d/dx Bx)^2 + (d/dx Cx)^2.
(7c) CC(x*y)^2 = (d/dy Ay)^2 + (d/dy By)^2 + (d/dy Cy)^2.
In the above, CC(x) and SS(x) are the e.g.f.s of A326551 and A326552 defined by
(8a) CC(x*y)^2 + SS(x*y)^2 = 1,
(8b) SS(x*y) = Integral CC(x*y) * (Integral CC(x*y) dy) dx,
(8c) CC(x*y) = 1 - Integral SS(x*y) * (Integral CC(x*y) dy) dx,
(8d) SS(x*y) = sin( Integral Integral CC(x*y) dx dy ),
(8e) CC(x*y) = cos( Integral Integral CC(x*y) dx dy ).
OTHER RELATIONS..
(9a) Ay = Ax*SS(x*y) - Bx*(d/dx Cx) + Cx*(d/dx Bx).
(9b) By = Bx*SS(x*y) - Cx*(d/dx Ax) + Ax*(d/dx Cx).
(9c) Cy = Cx*SS(x*y) - Ax*(d/dx Bx) + Bx*(d/dx Ax).
(9d) Ax = Ay*SS(x*y) - By*(d/dy Cy) + Cy*(d/dy By).
(9e) Bx = By*SS(x*y) - Cy*(d/dy Ay) + Ay*(d/dy Cy).
(9f) Cx = Cy*SS(x*y) - Ay*(d/dy By) + By*(d/dy Ay).
DERIVATIVES.
(10a) d/dx Ax = Bx*Cy - Cx*By.
(10b) d/dx Bx = Cx*Ay - Ax*Cy.
(10c) d/dx Cx = Ax*By - Bx*Ay.
(10d) d/dy Ay = By*Cx - Cy*Bx.
(10e) d/dy By = Cy*Ax - Ay*Cx.
(10f) d/dy Cy = Ay*Bx - By*Ax.
EXAMPLE
E.g.f.: B(x,y) = 1 + (-1*x^2)/2! + (1*x^4)/4! + (-1*x^6 + 120*x^4*y^2)/6! + (1*x^8 - 672*x^6*y^2)/8! + (-1*x^10 + 2160*x^8*y^2 - 120960*x^6*y^4)/10! + (1*x^12 - 5280*x^10*y^2 + 1584000*x^8*y^4)/12! + (-1*x^14 + 10920*x^12*y^2 - 10250240*x^10*y^4 + 482786304*x^8*y^6)/14! + (1*x^16 - 20160*x^14*y^2 + 45427200*x^12*y^4 - 11480268800*x^10*y^6)/16! + (-1*x^18 + 34272*x^16*y^2 - 157651200*x^14*y^4 + 124816613376*x^12*y^6 - 5405660282880*x^10*y^8)/18! + (1*x^20 - 54720*x^18*y^2 + 460158720*x^16*y^4 - 875447623680*x^14*y^6 + 203526629130240*x^12*y^8)/20! + ...
such that
. B(x,y) = 1 + Integral C(x,y)*A(y,x) - A(x,y)*C(y,x) dx,
. B(y,x) = 0 + Integral C(y,x)*A(x,y) - A(y,x)*C(x,y) dy,
where A(x,y) and C(x,y) satisfy
. A(x,y)^2 + B(x,y)^2 + C(x,y)^2 = 1.
TRIANGLE.
This triangle of coefficients T(n,k) of x^(2*n-2*k)*y^(2*k)/(2*n)! in B(x,y) begins
1;
-1;
1, 0;
-1, 120;
1, -672, 0;
-1, 2160, -120960;
1, -5280, 1584000, 0;
-1, 10920, -10250240, 482786304;
1, -20160, 45427200, -11480268800, 0;
-1, 34272, -157651200, 124816613376, -5405660282880;
1, -54720, 460158720, -875447623680, 203526629130240, 0;
-1, 83160, -1179763200, 4585597986816, -3340908170772480, 137550485329281024;
1, -121440, 2733857280, -19438470610944, 34039224247615488, -7523050148723687424, 0; ...
RELATED FUNCTIONS.
A(x,y) = x + (-1*x^3 - 3*x*y^2)/3! + (1*x^5 - 30*x^3*y^2 + 5*x*y^4)/5! + (-1*x^7 + 315*x^5*y^2 + 525*x^3*y^4 - 7*x*y^6)/7! + (1*x^9 - 1260*x^7*y^2 + 18270*x^5*y^4 - 2940*x^3*y^6 + 9*x*y^8)/9! + (-1*x^11 + 3465*x^9*y^2 - 496650*x^7*y^4 - 695310*x^5*y^6 + 10395*x^3*y^8 - 11*x*y^10)/11! + (1*x^13 - 7722*x^11*y^2 + 4279275*x^9*y^4 - 52144092*x^7*y^6 + 7702695*x^5*y^8 - 28314*x^3*y^10 + 13*x*y^12)/13! + (-1*x^15 + 15015*x^13*y^2 - 22387365*x^11*y^4 + 2701093395*x^9*y^6 + 3472834365*x^7*y^8 - 49252203*x^5*y^10 + 65065*x^3*y^12 - 15*x*y^14)/15! + ...
such that
. A(x,y) = 0 + Integral B(x,y)*C(y,x) - C(x,y)*B(y,x) dx,
. A(y,x) = 0 + Integral B(y,x)*C(x,y) - C(y,x)*B(x,y) dy.
C(x,y) = (2*x*y)/2! + (-4*x*y^3)/4! + (-160*x^3*y^3 + 6*x*y^5)/6! + (1344*x^3*y^5 - 8*x*y^7)/8! + (145152*x^5*y^5 - 5760*x^3*y^7 + 10*x*y^9)/10! + (-2534400*x^5*y^7 + 17600*x^3*y^9 - 12*x*y^11)/12! + (-551755776*x^7*y^7 + 20500480*x^5*y^9 - 43680*x^3*y^11 + 14*x*y^13)/14! + (16400384000*x^7*y^9 - 109025280*x^5*y^11 + 94080*x^3*y^13 - 16*x*y^15)/16! + ...
such that
. C(x,y) = 0 + Integral A(x,y)*B(y,x) - B(x,y)*A(y,x) dx,
. C(y,x) = 1 + Integral A(y,x)*B(x,y) - B(y,x)*A(x,y) dy.
CC(x) = 1 - 2*x^2/2!^2 + 56*x^4/4!^2 - 8336*x^6/6!^2 + 3985792*x^8/8!^2 - 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 - 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 + ... + A326551(n)*x^(2*n)/(2*n)!^2 + ...
such that
. CC(x*y) = sqrt( (Bx*Cy - Cx*By)^2 + (Cx*Ay - Ax*Cy)^2 + (Ax*By - Bx*Ay)^2 ).
SS(x) = x - 8*x^3/3!^2 + 576*x^5/5!^2 - 160768*x^7/7!^2 + 123535360*x^9/9!^2 - 212713734144*x^11/11!^2 + 716196297048064*x^13/13!^2 - 4280584942657732608*x^15/15!^2 + 42250703121584165486592*x^17/17!^2 + ... + A326552(n)*x^(2*n+1)/(2*n+1)!^2 + ...
such that SS(x*y) = Ax*Ay + Bx*By + Cx*Cy.
PROG
(PARI) {TBx(n, k) = my(Ax=x, Bx=1, Cx=x, Ay=y, By=y, Cy=1);
for(i=0, 2*n+1,
Ax = 0 + intformal( Bx*Cy - Cx*By, x) + O(x^(2*n+2));
Bx = 1 + intformal( Cx*Ay - Ax*Cy, x) + O(x^(2*n+2));
Cx = 0 + intformal( Ax*By - Bx*Ay, x) + O(x^(2*n+2));
Ay = 0 + intformal( By*Cx - Cy*Bx, y) + O(y^(2*n+2));
By = 0 + intformal( Cy*Ax - Ay*Cx, y) + O(y^(2*n+2));
Cy = 1 + intformal( Ay*Bx - By*Ax, y) + O(y^(2*n+2));
);
(2*n)! * polcoeff( polcoeff(Bx, 2*n-2*k, x), 2*k, y)}
for(n=0, 10, for(k=0, n\2, print1( TBx(n, k), ", ")); print(""))
CROSSREFS
Cf. A326797 (A), A326799 (C).
Cf. A326795 (row sums), A326551 (CC), A326552 (SS).
Sequence in context: A221406 A267428 A156739 * A090217 A267570 A267286
KEYWORD
sign,tabf
AUTHOR
Paul D. Hanna, Aug 03 2019
STATUS
approved

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Last modified May 27 13:56 EDT 2024. Contains 372861 sequences. (Running on oeis4.)