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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A048730 A307163 A254526

Adjacent sequences:  A322728 A322729 A322730 * A322732 A322733 A322734

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 26 2018

STATUS

approved

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Last modified July 8 07:16 EDT 2020. Contains 335513 sequences. (Running on oeis4.)