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A322730 E.g.f. S(x,y) = Integral C(x,y)*C(y,x) dx such that C(x,y)^2 - S(x,y)^2 = 1 and C(y,x) = 1 + Integral S(y,x)*C(x,y) dy, where S(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n+1-2*k)*y^(2*k)/(2*n+1)!, as a triangle of coefficients T(n,k) read by rows. 5
1, 1, 3, 1, 50, 5, 1, 693, 1155, 7, 1, 9972, 70686, 23268, 9, 1, 135575, 3479850, 4871790, 406725, 11, 1, 1727622, 157346475, 631853508, 283223655, 6334614, 13, 1, 20926185, 6802724565, 67722059405, 87071219235, 14965994043, 90680135, 15, 1, 243932456, 282646403340, 6596182917688, 19436510145910, 10365430299224, 734880648684, 1219662280, 17, 1, 2760372459, 11263126697316, 606536559381564, 3683652871295358, 4502242398249882, 1126425038851476, 33789380091948, 15642110601, 19 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

See A322220 for another description of the e.g.f. of this sequence:

T(n,k) = binomial(2*n+1,2*k) * A322220(n,k).

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. S(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^(2*n+1-2*k)*y^(2*k)/(2*n+1)! begins

S(x,y) = x + (1*x^3 + 3*x*y^2)/3! + (1*x^5 + 50*x^3*y^2 + 5*x*y^4)/5! + (1*x^7 + 693*x^5*y^2 + 1155*x^3*y^4 + 7*x*y^6)/7! + (1*x^9 + 9972*x^7*y^2 + 70686*x^5*y^4 + 23268*x^3*y^6 + 9*x*y^8)/9! + (1*x^11 + 135575*x^9*y^2 + 3479850*x^7*y^4 + 4871790*x^5*y^6 + 406725*x^3*y^8 + 11*x*y^10)/11! + (1*x^13 + 1727622*x^11*y^2 + 157346475*x^9*y^4 + 631853508*x^7*y^6 + 283223655*x^5*y^8 + 6334614*x^3*y^10 + 13*x*y^12)/13! + (1*x^15 + 20926185*x^13*y^2 + 6802724565*x^11*y^4 + 67722059405*x^9*y^6 + 87071219235*x^7*y^8 + 14965994043*x^5*y^10 + 90680135*x^3*y^12 + 15*x*y^14)/15! + ...

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.

TRIANGLE.

This triangle of coefficients T(n,k) of x^(2*n+1-2*k)*y^(2*k)/(2*n+1)! in S(x,y) starts

1;

1, 3;

1, 50, 5;

1, 693, 1155, 7;

1, 9972, 70686, 23268, 9;

1, 135575, 3479850, 4871790, 406725, 11;

1, 1727622, 157346475, 631853508, 283223655, 6334614, 13;

1, 20926185, 6802724565, 67722059405, 87071219235, 14965994043, 90680135, 15;

1, 243932456, 282646403340, 6596182917688, 19436510145910, 10365430299224, 734880648684, 1219662280, 17;

1, 2760372459, 11263126697316, 606536559381564, 3683652871295358, 4502242398249882, 1126425038851476, 33789380091948, 15642110601, 19; ...

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+1)! *polcoeff(polcoeff(Sx, 2*n+1-2*k, x), 2*k, y)}

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

CROSSREFS

Cf. A322731, A322732, A322733 (row sums).

Cf. A322220.

Sequence in context: A223173 A010292 A133104 * A292425 A095988 A189898

Adjacent sequences:  A322727 A322728 A322729 * A322731 A322732 A322733

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 26 2018

STATUS

approved

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Last modified July 8 00:40 EDT 2020. Contains 335502 sequences. (Running on oeis4.)