A326802 Consider the e.g.f. D(x,y) = sqrt(1/2) * Sum_{n>=0} Sum_{k=0..2*n} T(n,k) * x^(2*n-k) * y^k / ((2*n-k)!*k!) and related functions S(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<=2*n) of D(x,y).

1, -1, 1, 0, 1, 0, 0, -1, 0, -1, 0, 8, -8, 0, 1, 0, 1, 0, -24, 0, 0, 24, 0, -1, 0, -1, 0, 48, 0, -576, 576, 0, -48, 0, 1, 0, 1, 0, -80, 0, 3200, 0, 0, -3200, 0, 80, 0, -1, 0, -1, 0, 120, 0, -10240, 0, 160768, -160768, 0, 10240, 0, -120, 0, 1, 0, 1, 0, -168, 0, 24960, 0, -1433600, 0, 0, 1433600, 0, -24960, 0, 168, 0, -1, 0, -1, 0, 224, 0, -51520, 0, 6723584, 0, -123535360, 123535360, 0, -6723584, 0, 51520, 0, -224, 0, 1, 0, 1, 0, -288, 0, 94976, 0, -22586368, 0, 1615675392, 0, 0, -1615675392, 0, 22586368, 0, -94976, 0, 288, 0, -1, 0

Offset

0,12

Links

Paul D. Hanna, Table of n, a(n) for n = 0..3720 (the first 60 rows of the triangle).

Formula

The e.g.f. Dx = D(x,y) and related functions Sx = S(x,y), Cx = C(x,y), Sy = S(y,x), Cy = C(y,x), and Dy = D(y,x) satisfy the following relations.

DEFINITION.

(1a) Sx = Integral Cx*Dy + Cy*Dx dx,

(1b) Cx = sqrt(1/2) - Integral Sx*Dy + Sy*Dx dx,

(1c) Dx = sqrt(1/2) - Integral Sx*Cy - Sy*Cx dx,

(2a) Sy = Integral Cy*Dx + Cx*Dy dy,

(2b) Cy = sqrt(1/2) - Integral Sy*Dx + Sx*Dy dy,

(2c) Dy = sqrt(1/2) - Integral Sy*Cx - Sx*Cy dy.

IDENTITIES.

(3a) Dx^2 + Cx^2 + Sx^2 = 1.

(3b) Dy^2 + Cy^2 + Sy^2 = 1.

(4a) Dx*(d/dx Dx) + Cx*(d/dx Cx) + Sx*(d/dx Sx) = 0.

(4b) Dy*(d/dy Dy) + Cy*(d/dy Cy) + Sy*(d/dy Sy) = 0.

(4c) Dy*(d/dx Dx) - Cy*(d/dx Cx) - Sy*(d/dx Sx) = 0.

(4d) Dx*(d/dy Dy) - Cx*(d/dy Cy) - Sx*(d/dy Sy) = 0.

(5a) (Dx*Dy - Cx*Cy - Sx*Sy)^2 + (d/dx Dx)^2 + (d/dx Cx)^2 + (d/dx Sx)^2 = 1.

(5b) (Dx*Dy - Cx*Cy - Sx*Sy)^2 + (d/dy Dy)^2 + (d/dy Cy)^2 + (d/dy Sy)^2 = 1.

RELATED FUNCTIONS.

(6a) SS(x*y) = Dx*Dy - Cx*Cy - Sx*Sy.

(6b) d/dx SS(x*y) = Dx*(d/dx Dy) - Cx*(d/dx Cy) - Sx*(d/dx Sy).

(6c) d/dy SS(x*y) = Dy*(d/dy Dx) - Cy*(d/dy Cx) - Sy*(d/dy Sx).

(7a) CC(x*y)^2 = (Cx*Dy + Cy*Dx)^2 + (Sx*Dy + Sy*Dx)^2 + (Sx*Cy - Sy*Cx)^2.

(7b) CC(x*y)^2 = (d/dx Dx)^2 + (d/dx Cx)^2 + (d/dx Sx)^2.

(7c) CC(x*y)^2 = (d/dy Dy)^2 + (d/dy Cy)^2 + (d/dy Sy)^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 ).

DERIVATIVES.

(9a) d/dx Sx = Cx*Dy + Cy*Dx.

(9b) d/dx Cx = -Sx*Dy - Sy*Dx.

(9c) d/dx Dx = -Sx*Cy + Sy*Cx.

(9d) d/dy Sy = Sy*Dx + Sx*Dy.

(9e) d/dy Cy = -Sy*Dx - Sx*Dy.

(9f) d/dy Dy = -Sy*Cx + Sx*Cy.

Example

E.g.f.: D(x,y) = sqrt(1/2) * (1 + (-x^2/2! + x*y ) + ( x^4/4! - x*y^3/3! ) + (-x^6/6! + 8*x^4*y^2/(4!*2!) - 8*x^3*y^3/(3!*3!) + x*y^5/5! ) + ( x^8/8! - 24*x^6*y^2/(6!*2!) + 24*x^3*y^5/(3!*5!) - x*y^7/7! ) + (-x^10/10! + 48*x^8*y^2/(8!*2!) - 576*x^6*y^4/(6!*4!) + 576*x^5*y^5/(5!*5!) - 48*x^3*y^7/(3!*7!) + x*y^9/9! ) + ( x^12/12! - 80*x^10*y^2/(10!*2!) + 3200*x^8*y^4/(8!*4!) - 3200*x^5*y^7/(5!*7!) + 80*x^3*y^9/(3!*9!) - x*y^11/11! ) + (-x^14/14! + 120*x^12*y^2/(12!*2!) - 10240*x^10*y^4/(10!*4!) + 160768*x^8*y^6/(8!*6!) - 160768*x^7*y^7/(7!*7!) + 10240*x^5*y^9/(5!*9!) - 120*x^3*y^11/(3!*11!) + x*y^13/13! ) + ( x^16/16! - 168*x^14*y^2/(14!*2!) + 24960*x^12*y^4/(12!*4!) - 1433600*x^10*y^6/(10!*6!) + 1433600*x^7*y^9/(7!*9!) - 24960*x^5*y^11/(5!*11!) + 168*x^3*y^13/(3!*13!) - x*y^15/15! ) + (-1*x^18/18! + 224*x^16*y^2/(16!*2!) - 51520*x^14*y^4/(14!*4!) + 6723584*x^12*y^6/(12!*6!) - 123535360*x^10*y^8/(10!*8!) + 123535360*x^9*y^9/(9!*9!) - 6723584*x^7*y^11/(7!*11!) + 51520*x^5*y^13/(5!*13!) - 224*x^3*y^15/(3!*15!) + x*y^17/17! ) + ( x^20/20! - 288*x^18*y^2/(18!*2!) + 94976*x^16*y^4/(16!*4!) - 22586368*x^14*y^6/(14!*6!) + 1615675392*x^12*y^8/(12!*8!) - 1615675392*x^9*y^11/(9!*11!) + 22586368*x^7*y^13/(7!*13!) - 94976*x^5*y^15/(5!*15!) + 288*x^3*y^17/(3!*17!) - x*y^19/19! ) + ...).
This triangle of coefficients T(n,k) of x^(2*n-k)*y^k/((2*n-k)!*k!) in sqrt(2)*D(x,y) begins
1;
-1, 1, 0;
1, 0, 0, -1, 0;
-1, 0, 8, -8, 0, 1, 0;
1, 0, -24, 0, 0, 24, 0, -1, 0;
-1, 0, 48, 0, -576, 576, 0, -48, 0, 1, 0;
1, 0, -80, 0, 3200, 0, 0, -3200, 0, 80, 0, -1, 0;
-1, 0, 120, 0, -10240, 0, 160768, -160768, 0, 10240, 0, -120, 0, 1, 0;
1, 0, -168, 0, 24960, 0, -1433600, 0, 0, 1433600, 0, -24960, 0, 168, 0, -1, 0;
-1, 0, 224, 0, -51520, 0, 6723584, 0, -123535360, 123535360, 0, -6723584, 0, 51520, 0, -224, 0, 1, 0;
1, 0, -288, 0, 94976, 0, -22586368, 0, 1615675392, 0, 0, -1615675392, 0, 22586368, 0, -94976, 0, 288, 0, -1, 0;
-1, 0, 360, 0, -161280, 0, 61458432, 0, -10447847424, 0, 212713734144, -212713734144, 0, 10447847424, 0, -61458432, 0, 161280, 0, -360, 0, 1, 0;
1, 0, -440, 0, 257280, 0, -144420864, 0, 46282211328, 0, -3835832827904, 0, 0, 3835832827904, 0, -46282211328, 0, 144420864, 0, -257280, 0, 440, 0, -1, 0; ...
CENTRAL TERMS.
The central terms are found in 1 + SS(x*y) = 1 + Dx*Dy - Cx*Cy - Sx*Sy:
[1, 1, 0, -8, 0, 576, 0, -160768, 0, 123535360, 0, -212713734144, 0, 716196297048064, 0, -4280584942657732608, ...] (cf. A326552).
RELATED SERIES.
The e.g.f. of A326800 begins
S(x,y) = x + (-x^3/3! - x*y^2/2! ) + ( x^5/5! - 3*x^3*y^2/(3!*2!) + x*y^4/4! ) + (-x^7/7! + 15*x^5*y^2/(5!*2!) + 15*x^3*y^4/(3!*4!) - x*y^6/6! ) + ( x^9/9! - 35*x^7*y^2/(7!*2!) + 145*x^5*y^4/(5!*4!) - 35*x^3*y^6/(3!*6!) + x*y^8/8! ) + (-x^11/11! + 63*x^9*y^2/(9!*2!) - 1505*x^7*y^4/(7!*4!) - 1505*x^5*y^6/(5!*6!) + 63*x^3*y^8/(3!*8!) - x*y^10/10! ) + ( x^13/13! - 99*x^11*y^2/(11!*2!) + 5985*x^9*y^4/(9!*4!) - 30387*x^7*y^6/(7!*6!) + 5985*x^5*y^8/(5!*8!) - 99*x^3*y^10/(3!*10!) + x*y^12/12! ) + (-x^15/15! + 143*x^13*y^2/(13!*2!) - 16401*x^11*y^4/(11!*4!) + 539679*x^9*y^6/(9!*6!) + 539679*x^7*y^8/(7!*8!) - 16401*x^5*y^10/(5!*10!) + 143*x^3*y^12/(3!*12!) - x*y^14/14! ) + ...
The e.g.f. of A326801 begins
C(x,y) = sqrt(1/2) * (1 + (-x^2/2! - x*y ) + ( x^4/4! + x*y^3/3! ) + (-x^6/6! + 8*x^4*y^2/(4!*2!) + 8*x^3*y^3/(3!*3!) - x*y^5/5! ) + ( x^8/8! - 24*x^6*y^2/(6!*2!) - 24*x^3*y^5/(3!*5!) + x*y^7/7! ) + (-x^10/10! + 48*x^8*y^2/(8!*2!) - 576*x^6*y^4/(6!*4!) - 576*x^5*y^5/(5!*5!) + 48*x^3*y^7/(3!*7!) - x*y^9/9! ) + ( x^12/12! - 80*x^10*y^2/(10!*2!) + 3200*x^8*y^4/(8!*4!) + 3200*x^5*y^7/(5!*7!) - 80*x^3*y^9/(3!*9!) + x*y^11/11! ) + (-x^14/14! + 120*x^12*y^2/(12!*2!) - 10240*x^10*y^4/(10!*4!) + 160768*x^8*y^6/(8!*6!) + 160768*x^7*y^7/(7!*7!) - 10240*x^5*y^9/(5!*9!) + 120*x^3*y^11/(3!*11!) - x*y^13/13! ) + ...).
The e.g.f. of A326552 begins
SS(x*y) = (x*y) - 8*(x*y)^3/3!^2 + 576*(x*y)^5/5!^2 - 160768*(x*y)^7/7!^2 + 123535360*(x*y)^9/9!^2 - 212713734144*(x*y)^11/11!^2 + 716196297048064*(x*y)^13/13!^2 - 4280584942657732608*(x*y)^15/15!^2 + 42250703121584165486592*(x*y)^17/17!^2 - 651154631135458759089848320*(x*y)^19/19!^2 + 14983590319172065236171175755776*(x*y)^21/21!^2 + ... + A326552(n)*(x*y)^(2*n-1)/(2*n-1)! + ...
such that
SS(x*y) = Dx*Dy - Cx*Cy - Sx*Sy.
The e.g.f. of A326551 begins
CC(x*y) = 1 - 2*(x*y)^2/2!^2 + 56*(x*y)^4/4!^2 - 8336*(x*y)^6/6!^2 + 3985792*(x*y)^8/8!^2 - 4679517952*(x*y)^10/10!^2 + 11427218287616*(x*y)^12/12!^2 - 51793067942397952*(x*y)^14/14!^2 + 400951893341645930496*(x*y)^16/16!^2 - 4975999084909976839454720*(x*y)^18/18!^2 + 94178912073481319162642169856*(x*y)^20/20!^2 -+ ... + A326551(n)*(x*y)^(2*n)/(2*n)! + ...
such that
CC(x*y)^2 = (Cx*Dy + Cy*Dx)^2 + (Sx*Dy + Sy*Dx)^2 + (Sx*Cy - Sy*Cx)^2,
and CC(x*Y)^2 + SS(x*y)^2 = 1.

Prog

(PARI)
{TDx(n, k) = my(Cx=1, Sx=x, Dx=1, Cy=1, Sy=y, Dy=1);
for(i=0, 2*n+1,
Sx =             intformal( Cx*Dy + Cy*Dx, x) + O(x^(2*n+2));
Cx = sqrt(1/2) - intformal( Sx*Dy + Sy*Dx, x);
Dx = sqrt(1/2) - intformal( Sx*Cy - Sy*Cx, x);
Sy =             intformal( Cy*Dx + Cx*Dy, y) + O(y^(2*n+2));
Cy = sqrt(1/2) - intformal( Sy*Dx + Sx*Dy, y);
Dy = sqrt(1/2) - intformal( Sy*Cx - Sx*Cy, y);
);
round( (2*n-k)!*k! * polcoeff( polcoeff(sqrt(2)*Dx, 2*n-k, x), k, y) )}
for(n=0, 10, for(k=0, 2*n, print1( TDx(n, k), ", ")); print(""))

Crossrefs

Cf. A326800 (Sx), A326801 (Cx), A326551 (CC), A326552 (SS).

Sequence in context: A273818 A375369 A326801 * A179639 A113809 A231097

Adjacent sequences: A326799 A326800 A326801 * A326803 A326804 A326805

Keyword

sign,tabf

Author

Paul D. Hanna, Jul 27 2019

Status

approved