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.

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

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: A010292 A369756 A133104 * A292425 A095988 A189898

Adjacent sequences: A322727 A322728 A322729 * A322731 A322732 A322733

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 26 2018

Status

approved