0,4

Row reversal of triangle A324611.

Related identity: (1 + sin(z))/cos(z) = exp( Integral 1/cos(z) dz ).

Related identity: cos(x+y)*cos(x-y) = (1 - sin(x)^2 - sin(y)^2). - Paul D. Hanna, Sep 14 2024

Name changed Sep 14 2024; prior name was: E.g.f. C(x,y) = cosh( Integral C(x,y)*C(y,x) dx ), where C(y,x) = cosh( Integral C(x,y)*C(y,x) dy ).

Paul D. Hanna, Table of n, a(n) for n = 0..495 terms in rows 0..30 of this triangle in flattened form.

E.g.f. Cx = C(x,y) and related functions Sx = S(x,y), Cy = C(y,x), and Sy = S(y,x) satisfy the following relations.

(1a) Cx = 1 + Integral Sx * Cx*Cy dx.

(1b) Sx = Integral Cx * Cx*Cy dx.

(1c) Cy = 1 + Integral Sy * Cx*Cy dy.

(1d) Sy = Integral Cy * Cx*Cy dy.

(2a) Cx^2 - Sx^2 = 1.

(2b) Cy^2 - Sy^2 = 1.

(3a) Cx = cosh( Integral Cx*Cy dx ).

(3b) Sx = sinh( Integral Cx*Cy dx ).

(3c) Cy = cosh( Integral Cx*Cy dy ).

(3d) Sy = sinh( Integral Cx*Cy dy ).

(4a) Cx + Sx = exp( Integral Cx*Cy dx ).

(4b) Cy + Sy = exp( Integral Cx*Cy dy ).

(5a) (Cx + Sx)*(Cy + Sy) = (1 + sin(x+y))/cos(x+y).

(5b) (Cx + Sx)*(Cy - Sy) = (1 + sin(x-y))/cos(x-y).

(6a) Cx*Cy + Sx*Sy = 1/cos(x+y).

(6b) Cx*Sy + Sx*Cy = tan(x+y).

(7a) Cx*Cy = ( 1/cos(x+y) + 1/cos(x-y) )/2.

(7b) Sx*Sy = ( 1/cos(x+y) - 1/cos(x-y) )/2.

(7c) Cx*Sy = ( tan(x+y) - tan(x-y) )/2.

(7d) Sx*Cy = ( tan(x+y) + tan(x-y) )/2.

(8a) Cx*Cy = cos(x)*cos(y) / (cos(x+y)*cos(x-y)).

(8b) Sx*Sy = sin(x)*sin(y) / (cos(x+y)*cos(x-y)).

(8c) Cx*Sy = cos(y)*sin(y) / (cos(x+y)*cos(x-y)).

(8d) Sx*Cy = sin(x)*cos(x) / (cos(x+y)*cos(x-y)).

(9a) Cx + Sx = sqrt( (1 + sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ).

(9b) Cy + Sy = sqrt( (1 + sin(x+y))/cos(x+y) * (1 - sin(x-y))/cos(x-y) ).

(9c) Cx - Sx = sqrt( (1 - sin(x+y))/cos(x+y) * (1 - sin(x-y))/cos(x-y) ).

(9d) Cy - Sy = sqrt( (1 - sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ).

Let E(x,y) = sqrt( (1 + sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ) then

(10a) E(x,y) = C(x,y) + S(x,y) where E(-x,y) = 1/E(x,y),

(10b) C(x,y) = (E(x,y) + E(-x,y))/2,

(10c) S(x,y) = (E(x,y) - E(-x,y))/2.

From Paul D. Hanna, Sep 14 2024: (Start) Explicitly,

(11a) Cx = cos(y) / sqrt(1 - sin(x)^2 - sin(y)^2).

(11b) Sx = sin(x) / sqrt(1 - sin(x)^2 - sin(y)^2).

(11c) Cy = cos(x) / sqrt(1 - sin(x)^2 - sin(y)^2).

(11d) Sy = sin(y) / sqrt(1 - sin(x)^2 - sin(y)^2).

(End)

PARTICULAR ARGUMENTS.

E.g.f. at y = 0: C(x,y=0) = 1/cos(x).

E.g.f. at y = x: C(x,y=x) = cos(x)/sqrt(cos(2*x)).

FORMULAS INVOLVING TERMS.

T(n,0) = A000364(n) for n >= 0, where A000364 is the secant numbers.

T(n,n-1) = A000182(n) for n >= 1, where A000182 is the tangent numbers.

E.g.f.: C(x,y) = 1 + (1*x^2/2!) + (5*x^4/4! + 2*x^2*y^2/(2!*2!)) + (61*x^6/6! + 28*x^4*y^2/(4!*2!) + 16*x^2*y^4/(2!*4!)) + (1385*x^8/8! + 662*x^6*y^2/(6!*2!) + 440*x^4*y^4/(4!*4!) + 272*x^2*y^6/(2!*6!)) + (50521*x^10/10! + 24568*x^8*y^2/(8!*2!) + 17176*x^6*y^4/(6!*4!) + 12448*x^4*y^6/(4!*6!) + 7936*x^2*y^8/(2!*8!)) + (2702765*x^12/12! + 1326122*x^10*y^2/(10!*2!) + 949520*x^8*y^4/(8!*4!) + 727232*x^6*y^6/(6!*6!) + 546560*x^4*y^8/(4!*8!) + 353792*x^2*y^10/(2!*10!)) + ...

such that C(x,y) = cosh( Integral C(x,y)*C(y,x) dx ).

Explicitly,

C(x,y) = ( sqrt( (1 + sin(x+y))/cos(x+y) * (1 + sin(x-y))/cos(x-y) ) + sqrt( (1 - sin(x+y))/cos(x+y) * (1 - sin(x-y))/cos(x-y) ) )/2.

This triangle of coefficients T(n,k) of x^(2*n-2*k)*y^(2*k)/((2*n-2*k)!*(2*k)!) in e.g.f. C(x,y) begins

1;

1, 0;

5, 2, 0;

61, 28, 16, 0;

1385, 662, 440, 272, 0;

50521, 24568, 17176, 12448, 7936, 0;

2702765, 1326122, 949520, 727232, 546560, 353792, 0;

199360981, 98329108, 71350336, 56140288, 44720896, 34259968, 22368256, 0;

19391512145, 9596075582, 7020926600, 5610570992, 4600173440, 3742967552, 2900372480, 1903757312, 0;

2404879675441, 1192744081648, 877465887496, 708137588128, 590470281856, 495154244608, 408133590016, 318605529088, 209865342976, 0; ...

RELATED SERIES.

C(y,x) = 1 + (1*y^2/2!) + (2*x^2*y^2/(2!*2!) + 5*y^4/4!) + (16*x^4*y^2/(4!*2!) + 28*x^2*y^4/(2!*4!) + 61*y^6/6!) + (272*x^6*y^2/(6!*2!) + 440*x^4*y^4/(4!*4!) + 662*x^2*y^6/(2!*6!) + 1385*y^8/8!) + (7936*x^8*y^2/(8!*2!) + 12448*x^6*y^4/(6!*4!) + 17176*x^4*y^6/(4!*6!) + 24568*x^2*y^8/(2!*8!) + 50521*y^10/10!) + (353792*x^10*y^2/(10!*2!) + 546560*x^8*y^4/(8!*4!) + 727232*x^6*y^6/(6!*6!) + 949520*x^4*y^8/(4!*8!) + 1326122*x^2*y^10/(2!*10!) + 2702765*y^12/12!) + ...

such that C(y,x) = cosh( Integral C(x,y)*C(y,x) dy ).

S(x,y) = x + (2*x^3/3! + 1*x*y^2/2!) + (16*x^5/5! + 8*x^3*y^2/(3!*2!) + 5*x*y^4/4!) + (272*x^7/7! + 136*x^5*y^2/(5!*2!) + 94*x^3*y^4/(3!*4!) + 61*x*y^6/6!) + (7936*x^9/9! + 3968*x^7*y^2/(7!*2!) + 2840*x^5*y^4/(5!*4!) + 2108*x^3*y^6/(3!*6!) + 1385*x*y^8/8!) + (353792*x^11/11! + 176896*x^9*y^2/(9!*2!) + 128704*x^7*y^4/(7!*4!) + 100096*x^5*y^6/(5!*6!) + 76474*x^3*y^8/(3!*8!) + 50521*x*y^10/10!) + (22368256*x^13/13! + 11184128*x^11*y^2/(11!*2!) + 8211200*x^9*y^4/(9!*4!) + 6531968*x^7*y^6/(7!*6!) + 5261120*x^5*y^8/(5!*8!) + 4079408*x^3*y^10/(3!*10!) + 2702765*x*y^12/12!) + ...

such that C(x,y)^2 - S(x,y)^2 = 1.

S(y,x) = y + (1*x^2*y/2! + 2*y^3/3!) + (5*x^4*y/4! + 8*x^2*y^3/(2!*3!) + 16*y^5/5!) + (61*x^6*y/6! + 94*x^4*y^3/(4!*3!) + 136*x^2*y^5/(2!*5!) + 272*y^7/7!) + (1385*x^8*y/8! + 2108*x^6*y^3/(6!*3!) + 2840*x^4*y^5/(4!*5!) + 3968*x^2*y^7/(2!*7!) + 7936*y^9/9!) + (50521*x^10*y/10! + 76474*x^8*y^3/(8!*3!) + 100096*x^6*y^5/(6!*5!) + 128704*x^4*y^7/(4!*7!) + 176896*x^2*y^9/(2!*9!) + 353792*y^11/11!) + (2702765*x^12*y/12! + 4079408*x^10*y^3/(10!*3!) + 5261120*x^8*y^5/(8!*5!) + 6531968*x^6*y^7/(6!*7!) + 8211200*x^4*y^9/(4!*9!) + 11184128*x^2*y^11/(2!*11!) + 22368256*y^13/13!) + ...

such that C(y,x)^2 - S(y,x)^2 = 1.

(PARI) {T(n, k) = my(Cx = 1 + x*O(x^(2*n)), Cy = 1 + y*O(y^(2*n)));

for(i=1, 2*n,

Cx = cosh(intformal(Cx*Cy, x));

Cy = cosh(intformal(Cx*Cy, y)); );

Sx = sinh(intformal(Cx*Cy, x));

Sy = sinh(intformal(Cx*Cy, y));

(2*n-2*k)!*(2*k)! * polcoeff(polcoeff(Cx, 2*n-2*k, x), 2*k, y)}

for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))

Cf. A324610 (S(x,y)), A324611 (C(y,x)), A324612 (S(y,x)).

Cf. A000364 (T(n,0)), A000182 (T(n,n-1)).

Cf. A322221 (variant).

Sequence in context: A058512 A111560 A371026 * A211991 A354596 A303127

Adjacent sequences: A324606 A324607 A324608 * A324610 A324611 A324612

nonn,tabl

Paul D. Hanna, Mar 09 2019

approved