A322732 E.g.f. C(y,x) = 1 + Integral S(y,x)*C(x,y) dy such that C(x,y)^2 - S(x,y)^2 = 1 and C(x,y) = Integral S(x,y)*C(y,x) dx, where C(y,x) = 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, 0, 1, 0, 12, 1, 0, 120, 180, 1, 0, 896, 9520, 2632, 1, 0, 5760, 369600, 504000, 37080, 1, 0, 33792, 12735360, 57376704, 23562000, 487476, 1, 0, 186368, 399246848, 5256451200, 6645999360, 1039654616, 6045676, 1, 0, 983040, 11498905600, 427859367936, 1336544352000, 674286412800, 44074736160, 71745360, 1, 0, 5013504, 308405698560, 32219490680832, 224718683904000, 271641150443520, 63624274826112, 1793634471360, 823269744, 1

Offset

0,5

Comments

See A322222 for another description of the e.g.f. of this sequence:

T(n,k) = binomial(2*n,2*k) * A322222(n,k).

Row reversal of A322731.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1274 terms of this triangle read by 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(y,x) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n-2*k)*y^(2*k)/(2*n)! begins
C(y,x) = 1 + (1*y^2)/2! + (12*x^2*y^2 + 1*y^4)/4! + (120*x^4*y^2 + 180*x^2*y^4 + 1*y^6)/6! + (896*x^6*y^2 + 9520*x^4*y^4 + 2632*x^2*y^6 + 1*y^8)/8! + (5760*x^8*y^2 + 369600*x^6*y^4 + 504000*x^4*y^6 + 37080*x^2*y^8 + 1*y^10)/10! + (33792*x^10*y^2 + 12735360*x^8*y^4 + 57376704*x^6*y^6 + 23562000*x^4*y^8 + 487476*x^2*y^10 + 1*y^12)/12! + (186368*x^12*y^2 + 399246848*x^10*y^4 + 5256451200*x^8*y^6 + 6645999360*x^6*y^8 + 1039654616*x^4*y^10 + 6045676*x^2*y^12 + 1*y^14)/14! + (983040*x^14*y^2 + 11498905600*x^12*y^4 + 427859367936*x^10*y^6 + 1336544352000*x^8*y^8 + 674286412800*x^6*y^10 + 44074736160*x^4*y^12 + 71745360*x^2*y^14 + 1*y^16)/16! + ...
This series may be defined by
C(y,x) = cosh( Integral C(x,y) dy ), and
C(x,y) = cosh( Integral C(y,x) dx ).
TRIANGLE.
This triangle of coefficients T(n,k) of x^(2*n-2*k)*y^(2*k)/(2*n)! in C(y,x) starts
1;
0, 1;
0, 12, 1;
0, 120, 180, 1;
0, 896, 9520, 2632, 1;
0, 5760, 369600, 504000, 37080, 1;
0, 33792, 12735360, 57376704, 23562000, 487476, 1;
0, 186368, 399246848, 5256451200, 6645999360, 1039654616, 6045676, 1;
0, 983040, 11498905600, 427859367936, 1336544352000, 674286412800, 44074736160, 71745360, 1;
0, 5013504, 308405698560, 32219490680832, 224718683904000, 271641150443520, 63624274826112, 1793634471360, 823269744, 1; ...

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

Crossrefs

Cf. A322730, A322731.

Cf. A322222.

Sequence in context: A012453 A284453 A012454 * A370332 A370432 A325827

Adjacent sequences: A322729 A322730 A322731 * A322733 A322734 A322735

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 01 2019

Status

approved