A322731 E.g.f. C(x,y) = 1 + Integral S(x,y)*C(y,x) dx such that C(x,y)^2 - S(x,y)^2 = 1 and C(y,x) = Integral S(y,x)*C(x,y) dy, where C(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k)*y^(2*k)/(2*n)!, as a triangle of coefficients T(n,k) read by rows.

1, 1, 0, 1, 12, 0, 1, 180, 120, 0, 1, 2632, 9520, 896, 0, 1, 37080, 504000, 369600, 5760, 0, 1, 487476, 23562000, 57376704, 12735360, 33792, 0, 1, 6045676, 1039654616, 6645999360, 5256451200, 399246848, 186368, 0, 1, 71745360, 44074736160, 674286412800, 1336544352000, 427859367936, 11498905600, 983040, 0, 1, 823269744, 1793634471360, 63624274826112, 271641150443520, 224718683904000, 32219490680832, 308405698560, 5013504, 0

Offset

0,5

Comments

See A322221 for another description of the e.g.f. of this sequence:

T(n,k) = binomial(2*n,2*k) * A322221(n,k).

Row reversal of A322732.

Links

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

Formula

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.

Example

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)! begins
C(x,y) = 1 + (1*x^2)/2! + (1*x^4 + 12*x^2*y^2)/4! + (1*x^6 + 180*x^4*y^2 + 120*x^2*y^4)/6! + (1*x^8 + 2632*x^6*y^2 + 9520*x^4*y^4 + 896*x^2*y^6)/8! + (1*x^10 + 37080*x^8*y^2 + 504000*x^6*y^4 + 369600*x^4*y^6 + 5760*x^2*y^8)/10! + (1*x^12 + 487476*x^10*y^2 + 23562000*x^8*y^4 + 57376704*x^6*y^6 + 12735360*x^4*y^8 + 33792*x^2*y^10)/12! + (1*x^14 + 6045676*x^12*y^2 + 1039654616*x^10*y^4 + 6645999360*x^8*y^6 + 5256451200*x^6*y^8 + 399246848*x^4*y^10 + 186368*x^2*y^12)/14! + (1*x^16 + 71745360*x^14*y^2 + 44074736160*x^12*y^4 + 674286412800*x^10*y^6 + 1336544352000*x^8*y^8 + 427859367936*x^6*y^10 + 11498905600*x^4*y^12 + 983040*x^2*y^14)/16! + ...
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)! in C(x,y) starts
1;
1, 0;
1, 12, 0;
1, 180, 120, 0;
1, 2632, 9520, 896, 0;
1, 37080, 504000, 369600, 5760, 0;
1, 487476, 23562000, 57376704, 12735360, 33792, 0;
1, 6045676, 1039654616, 6645999360, 5256451200, 399246848, 186368, 0;
1, 71745360, 44074736160, 674286412800, 1336544352000, 427859367936, 11498905600, 983040, 0;
1, 823269744, 1793634471360, 63624274826112, 271641150443520, 224718683904000, 32219490680832, 308405698560, 5013504, 0; ...

Prog

(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)! *polcoeff(polcoeff(Cx, 2*n-2*k, x), 2*k, y)}
for(n=0, 10, for(k=0, n, print1( T(n, k), ", ")); print(""))

Crossrefs

Cf. A322730, A322732, A322734 (row sums).

Cf. A322221.

Sequence in context: A246223 A199542 A304330 * A370330 A370430 A113923

Adjacent sequences: A322728 A322729 A322730 * A322732 A322733 A322734

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 26 2018

Status

approved