0,5

See A322731 for another description of the e.g.f. of this sequence:

T(n,k) = A322731(n,k)/binomial(2*n,2*k).

Paul D. Hanna, Table of n, a(n) for n = 0..1274 terms of this triangle as read by rows 0..50

The special functions S(x,y), C(x,y), and D(x,y) satisfy the following relations.

(1a) S(x,y) = Integral C(x,y) * C(y,x) dx.

(1b) S(y,x) = Integral C(y,x) * C(x,y) dy.

(1c) C(x,y) = 1 + Integral S(x,y) * C(y,x) dx.

(1d) C(y,x) = 1 + Integral S(y,x) * C(x,y) dy.

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

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

(3a) S(x,y) = sinh( Integral C(y,x) dx ).

(3b) S(y,x) = sinh( Integral C(x,y) dy ).

(3c) C(x,y) = cosh( Integral C(y,x) dx ).

(3d) C(y,x) = cosh( Integral C(x,y) dy ).

(4a) C(x,y) + S(x,y) = exp( Integral C(y,x) dx ).

(4b) C(y,x) + S(y,x) = exp( Integral C(x,y) dy ).

(5a) d/dx S(x,y) = C(x,y) * C(y,x).

(5b) d/dx C(x,y) = S(x,y) * C(y,x).

(5c) d/dy S(y,x) = C(y,x) * C(x,y).

(5d) d/dy C(y,x) = S(y,x) * C(x,y).

Introducing function D(x,y) completes the symmetric relations as follows.

(6a) D(x,y) = Integral S(y,x) * C(x,y) dx.

(6b) D(y,x) = Integral S(x,y) * C(y,x) dy.

(7a) S(x,y) = sinh(x) + Integral C(x,y) * D(x,y) dy.

(7b) S(y,x) = sinh(y) + Integral C(y,x) * D(y,x) dx.

(7c) C(x,y) = cosh(x) + Integral S(x,y) * D(x,y) dy.

(7d) C(y,x) = cosh(y) + Integral S(y,x) * D(y,x) dx.

(8a) C(x,y) + S(x,y) = exp( x + Integral D(x,y) dy ).

(8b) C(y,x) + S(y,x) = exp( y + Integral D(y,x) dx ).

(9a) Integral C(y,x) dx = x + Integral D(x,y) dy.

(9b) Integral C(x,y) dy = y + Integral D(y,x) dx.

(10a) d/dy S(x,y) = C(x,y) * D(x,y).

(10b) d/dy C(x,y) = S(x,y) * D(x,y).

(10c) d/dx S(y,x) = C(y,x) * D(y,x).

(10d) d/dx C(y,x) = S(y,x) * D(y,x).

(10e) d/dx D(x,y) = S(y,x) * C(x,y).

(10f) d/dy D(y,x) = S(x,y) * C(y,x).

For brevity, let Cx = C(x,y), Cy = C(y,x), Sx = S(x,y), Sy = S(y,x), Dx = D(x,y), Dy = D(y,x), then further relations may be written as follows.

(11a) Cx*Cy + Sx*Sy = cosh(y) + Integral (Cy + Dy)*(Sx*Cy + Cx*Sy) dx.

(11b) Sx*Cy + Cx*Sy = sinh(y) + Integral (Cy + Dy)*(Cx*Cy + Sx*Sy) dx.

(11c) Cx*Cy + Sx*Sy = cosh(x) + Integral (Cx + Dx)*(Sx*Cy + Cx*Sy) dy.

(11d) Sx*Cy + Cx*Sy = sinh(x) + Integral (Cx + Dx)*(Cx*Cy + Sx*Sy) dy.

(12a) (Cx + Sx)*(Cy + Sy) = exp( y + Integral Cy + Dy dx ).

(12b) (Cx + Sx)*(Cy + Sy) = exp( x + Integral Cx + Dx dy ).

(12c) (Cx + Sx)*(Cy + Sy) = exp( x + y + Integral Dx dy + Integral Dy dx ).

(12d) (Cx + Sx)*(Cy + Sy) = exp( x + y + Integral Integral Sx*Cy + Cx*Sy dx dy ).

(12e) x + Integral (Cx + Dx) dy = y + Integral (Cy + Dy) dx.

(13a) d/dx (Cx + Sx)*(Cy + Sy) = (Cx + Sx)*(Cy + Sy)*(Cy + Dy).

(13b) d/dy (Cx + Sx)*(Cy + Sy) = (Cx + Sx)*(Cy + Sy)*(Cx + Dx).

(14a) (Cx + Sx)*(Cy + Sy) = exp(y) + Integral (Cx + Sx)*(Cy + Sy)*(Cy + Dy) dx.

(14b) (Cx + Sx)*(Cy + Sy) = exp(x) + Integral (Cx + Sx)*(Cy + Sy)*(Cx + Dx) dy.

E.g.f. C(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k)*y^(2*k)/((2*n-2*k)!*(2*k)!) begins

C(x,y) = 1 + 1*x^2/2! + (1*x^4/4! + 2*x^2*y^2/(2!*2!)) + (1*x^6/6! + 12*x^4*y^2/(4!*2!) + 8*x^2*y^4/(2!*4!)) + (1*x^8/8! + 94*x^6*y^2/(6!*2!) + 136*x^4*y^4/(4!*4!) + 32*x^2*y^6/(2!*6!)) + (1*x^10/10! + 824*x^8*y^2/(8!*2!) + 2400*x^6*y^4/(6!*4!) + 1760*x^4*y^6/(4!*6!) + 128*x^2*y^8/(2!*8!)) + (1*x^12/12! + 7386*x^10*y^2/(10!*2!) + 47600*x^8*y^4/(8!*4!) + 62096*x^6*y^6/(6!*6!) + 25728*x^4*y^8/(4!*8!) + 512*x^2*y^10/(2!*10!)) + (1*x^14/14! + 66436*x^12*y^2/(12!*2!) + 1038616*x^10*y^4/(10!*4!) + 2213120*x^8*y^6/(8!*6!) + 1750400*x^6*y^8/(6!*8!) + 398848*x^4*y^10/(4!*10!) + 2048*x^2*y^12/(2!*12!)) + (1*x^16/16! + 597878*x^14*y^2/(14!*2!) + 24216888*x^12*y^4/(12!*4!) + 84201600*x^10*y^6/(10!*6!) + 103849600*x^8*y^8/(8!*8!) + 53428992*x^6*y^10/(6!*10!) + 6318080*x^4*y^12/(4!*12!) + 8192*x^2*y^14/(2!*14!)) + ...

This series may be defined by

C(x,y) = cosh( Integral C(y,x) dx ), and

C(y,x) = cosh( Integral C(x,y) dy ).

TRIANGLE.

This triangle of coefficients T(n,k) of x^(2*n-2*k)*y^(2*k)/((2*n-2*k)!*(2*k)!) in C(x,y) starts

1;

1, 0;

1, 2, 0;

1, 12, 8, 0;

1, 94, 136, 32, 0;

1, 824, 2400, 1760, 128, 0;

1, 7386, 47600, 62096, 25728, 512, 0;

1, 66436, 1038616, 2213120, 1750400, 398848, 2048, 0;

1, 597878, 24216888, 84201600, 103849600, 53428992, 6318080, 8192, 0;

1, 5380848, 586155056, 3427293408, 6207805440, 5135488000, 1735589888, 100786176, 32768, 0;

1, 48427570, 14448604000, 147664947312, 387334142976, 460536301312, 268962125824, 58877726720, 1611169792, 131072, 0; ...

RELATED SERIES.

The related series S(x,y), where C(x,y)^2 - S(x,y)^2 = 1, begins

S(x,y) = x + (1*x^3/3! + 1*x*y^2/2!) + (1*x^5/5! + 5*x^3*y^2/(3!*2!) + 1*x*y^4/4!) + (1*x^7/7! + 33*x^5*y^2/(5!*2!) + 33*x^3*y^4/(3!*4!) + 1*x*y^6/6!) + (1*x^9/9! + 277*x^7*y^2/(7!*2!) + 561*x^5*y^4/(5!*4!) + 277*x^3*y^6/(3!*6!) + 1*x*y^8/8!) + (1*x^11/11! + 2465*x^9*y^2/(9!*2!) + 10545*x^7*y^4/(7!*4!) + 10545*x^5*y^6/(5!*6!) + 2465*x^3*y^8/(3!*8!) + 1*x*y^10/10!) + (1*x^13/13! + 22149*x^11*y^2/(11!*2!) + 220065*x^9*y^4/(9!*4!) + 368213*x^7*y^6/(7!*6!) + 220065*x^5*y^8/(5!*8!) + 22149*x^3*y^10/(3!*10!) +1*x*y^12/12!) + (1*x^15/15! + 199297*x^13*y^2/(13!*2!) + 4983681*x^11*y^4/(11!*4!) + 13530881*x^9*y^6/(9!*6!) + 13530881*x^7*y^8/(7!*8!) + 4983681*x^5*y^10/(5!*10!) + 199297*x^3*y^12/(3!*12!) + 1*x*y^14/14!) + ...

The series S(x,y) may be defined by

S(x,y) = Integral C(x,y) * C(y,x) dx, and

S(y,x) = Integral C(y,x) * C(x,y) dy,

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

The related series D(x,y) = Integral S(y,x) * C(x,y) dx, begins

D(x,y) = x*y + (2*x^3*y/3! + 1*x*y^3/3!) + (8*x^5*y/5! + 12*x^3*y^3/(3!*3!) + 1*x*y^5/5!) + (32*x^7*y/7! + 136*x^5*y^3/(5!*3!) + 94*x^3*y^5/(3!*5!) + 1*x*y^7/7!) + (128*x^9*y/9! + 1760*x^7*y^3/(7!*3!) + 2400*x^5*y^5/(5!*5!) + 824*x^3*y^7/(3!*7!) + 1*x*y^9/9!) + (512*x^11*y/11! + 25728*x^9*y^3/(9!*3!) + 62096*x^7*y^5/(7!*5!) + 47600*x^5*y^7/(5!*7!) + 7386*x^3*y^9/(3!*9!) + 1*x*y^11/11!) + (2048*x^13*y/13! + 398848*x^11*y^3/(11!*3!) + 1750400*x^9*y^5/(9!*5!) + 2213120*x^7*y^7/(7!*7!) + 1038616*x^5*y^9/(5!*9!) + 66436*x^3*y^11/(3!*11!) + 1*x*y^13/13!) + (8192*x^15*y/15! + 6318080*x^13*y^3/(13!*3!) + 53428992*x^11*y^5/(11!*5!) + 103849600*x^9*y^7/(9!*7!) + 84201600*x^7*y^9/(7!*9!) + 24216888*x^5*y^11/(5!*11!) + 597878*x^3*y^13/(3!*13!) + 1*x*y^15/15!) + ...

nmax = 10; m = 2 nmax - 1; s[x_, _] = x; s[y_, _] = y; c[_, _] = 1; Do[ s[x_, y_] = Integrate[Series[c[x, y] c[y, x], {x, 0, m}, {y, 0, m}], x] // Normal; c[x_, y_] = 1+Integrate[Series[s[x, y] c[y, x], {x, 0, m}, {y, 0, m}], x] // Normal; s[y_, x_] = Integrate[Series[c[y, x] c[x, y], {x, 0, m}, {y, 0, m}], y] // Normal; c[y_, x_] = 1+Integrate[Series[s[y, x] c[x, y], {x, 0, m}, {y, 0, m}], y] // Normal; Print[c[x, y] // Short], {nmax}];

T[n_, k_] := (2n-2k)! (2k)! SeriesCoefficient[c[x, y], {x, 0, 2n-2k}, {y, 0, 2k}];

Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 06 2019, from PARI *)

(PARI) {T(n, k) = my(Sx=x, Sy=y, Cx=1, Cy=1); for(i=1, 2*n,

Sx = intformal( Cx*Cy +x*O(x^(2*n)), x);

Cx = 1 + intformal( Sx*Cy, x);

Sy = intformal( Cy*Cx +y*O(y^(2*k)), y);

Cy = 1 + intformal( Sy*Cx, 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. A322220, A322222, A322223, A322224.

Cf. A322731.

Cf. A324609 (variant).

Sequence in context: A361951 A137514 A367381 * A328924 A217654 A267164

Adjacent sequences: A322218 A322219 A322220 * A322222 A322223 A322224

nonn,tabl

Paul D. Hanna, Dec 23 2018

approved