A322193 E.g.f.: C(x,y) = cosh(x)*cosh(y) / (1 - sinh(x)*sinh(y)), where C(x,y) = Sum_{n>=0} Sum_{k=0..2*n} T(n,k) * x^(2*n-k)*y^k/((2*n-k)!*k!), as a triangle of coefficients T(n,k) read by rows.

1, 1, 1, 1, 1, 4, 5, 4, 1, 1, 16, 41, 52, 41, 16, 1, 1, 64, 365, 784, 977, 784, 365, 64, 1, 1, 256, 3281, 12352, 23801, 29056, 23801, 12352, 3281, 256, 1, 1, 1024, 29525, 196864, 589217, 1049344, 1257125, 1049344, 589217, 196864, 29525, 1024, 1, 1, 4096, 265721, 3146752, 14677961, 37789696, 63318641, 74628352, 63318641, 37789696, 14677961, 3146752, 265721, 4096, 1, 1, 16384, 2391485, 50335744, 366476657, 1360482304, 3140590685, 5010663424, 5823720257, 5010663424, 3140590685, 1360482304, 366476657, 50335744, 2391485, 16384, 1

Offset: 0

Author

Paul D. Hanna, Dec 20 2018

Keywords

• nonn
• tabf

Comments

See A322621 for another description of the e.g.f. of this sequence.

Examples

			E.g.f.: C(x,y) = 1 + (1*x^2/2! + 1*x*y + 1*y^2/2!) + (1*x^4/4! + 4*x^3*y/3! + 5*x^2*y^2/(2!*2!) + 4*x*y^3/3! + 1*y^4/4!) + (1*x^6/6! + 16*x^5*y/5! + 41*x^4*y^2/(4!*2!) + 52*x^3*y^3/(3!*3!) + 41*x^2*y^4/(2!*4!) + 16*x*y^5/5! + 1*y^6/6!) + (1*x^8/8! + 64*x^7*y/7! + 365*x^6*y^2/(6!*2!) + 784*x^5*y^3/(5!*3!) + 977*x^4*y^4/(4!*4!) + 784*x^3*y^5/(3!*5!) + 365*x^2*y^6/(2!*6!) + 64*x*y^7/7! + 1*y^8/8!) + ...
where C(x,y) = cosh(x)*cosh(y) / (1 - sinh(x)*sinh(y)).
This irregular triangle of coefficients of x^(2*n-k)*y^k/((2*n-k)!*k!) in C(x,y) begins
1;
1, 1, 1;
1, 4, 5, 4, 1;
1, 16, 41, 52, 41, 16, 1;
1, 64, 365, 784, 977, 784, 365, 64, 1;
1, 256, 3281, 12352, 23801, 29056, 23801, 12352, 3281, 256, 1;
1, 1024, 29525, 196864, 589217, 1049344, 1257125, 1049344, 589217, 196864, 29525, 1024, 1;
1, 4096, 265721, 3146752, 14677961, 37789696, 63318641, 74628352, 63318641; 37789696, 14677961, 3146752, 265721, 4096, 1; ...
RELATED SERIES.
The series S(x,y), such that C(x,y)^2 - S(x,y)^2 = 1, begins
S(x,y) = (1*x + 1*y) + (1*x^3/3! + 2*x^2*y/2! + 2*x*y^2/2! + 1*y^3/3!) + (1*x^5/5! + 8*x^4*y/4! + 14*x^3*y^2/(3!*2!) + 14*x^2*y^3/(2!*3!) + 8*x*y^4/4! + 1*y^5/5!) + (1*x^7/7! + 32*x^6*y/6! + 122*x^5*y^2/(5!*2!) + 200*x^4*y^3/(4!*3!) + 200*x^3*y^4/(3!*4!) + 122*x^2*y^5/(2!*5!) + 32*x*y^6/6! + 1*y^7/7!) + ...
The e.g.f. may be written with coefficients of x^(2*n-k)*y^k/(2*n)!, as follows:
C(x,y) = 1 + (1*x^2 + 2*x*y + 1*y^2)/2! + (1*x^4 + 16*x^3*y + 30*x^2*y^2 + 16*x*y^3 + 1*y^4)/4! + (1*x^6 + 96*x^5*y + 615*x^4*y^2 + 1040*x^3*y^3 + 615*x^2*y^4 + 96*x*y^5 + 1*y^6)/6! + (1*x^8 + 512*x^7*y + 10220*x^6*y^2 + 43904*x^5*y^3 + 68390*x^4*y^4 + 43904*x^3*y^5 + 10220*x^2*y^6 + 512*x*y^7 + 1*y^8)/8! + ...
these coefficients are described by triangle A322621.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..2600 terms of this triangle as read by rows 0..50.

Crossrefs

Cf. A322190 (C + S), A322194 (S), A322195 (main diagonal).

Programs

Mathematica

T[n_, k_] := (2n-k)! k! SeriesCoefficient[Cosh[x] Cosh[y]/(1-Sinh[x] Sinh[y]), {x, 0, 2n-k}, {y, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, 2n}] // Flatten (* Jean-François Alcover, Dec 29 2018 *)

PARI

{T(n, k) = my(X=x+x*O(x^(2*n-k)), Y=y+y*O(y^k));
C = cosh(X)*cosh(Y)/(1 - sinh(X)*sinh(Y));
(2*n-k)!*k!*polcoeff(polcoeff(C, 2*n-k, x), k, y)}
/* Print as a triangle */
for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))

Formula

E.g.f.: C(x,y) and related series S(x,y) satisfy the following identities.

(1) C(x,y)^2 - S(x,y)^2 = 1.

(2a) C(x,y) = cosh(x) * cosh(y) / (1 - sinh(x)*sinh(y)).

(2b) S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).

(3a) cosh(x) = C(x,y) * cosh(y) / (1 + sinh(y)*S(x,y)).

(3b) sinh(x) = (S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).

(3c) cosh(y) = C(x,y) * cosh(x) / (1 + sinh(x)*S(x,y)).

(3d) sinh(y) = (S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).

(4a) exp(x) = (C(x,y)*cosh(y) + S(x,y) - sinh(y)) / (1 + sinh(y)*S(x,y)).

(4b) exp(y) = (C(x,y)*cosh(x) + S(x,y) - sinh(x)) / (1 + sinh(x)*S(x,y)).

(5a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) * (cosh(y) - sinh(y)*C(x,y)) / (1 - sinh(y)^2*S(x,y)^2).

(5b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) * (cosh(x) - sinh(x)*C(x,y)) / (1 - sinh(x)^2*S(x,y)^2).

(5c) C(x,y) + S(x,y) = (cosh(x) + sinh(x)*cosh(y)) * (cosh(y) + sinh(y)*cosh(x)) / (1 - sinh(x)^2*sinh(y)^2).

(6a) exp(x) = (C(x,y) + S(x,y)*cosh(y)) / (cosh(y) + sinh(y)*C(x,y)).

(6b) exp(y) = (C(x,y) + S(x,y)*cosh(x)) / (cosh(x) + sinh(x)*C(x,y)).

(6c) C(x,y) + S(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)).

(6d) C(x,y) + S(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)).

SPECIAL ARGUMENTS.

C(x, y=0) = cosh(x).

C(x, y=x) = cosh(x)^2 / (1 - sinh(x)^2).

C(x, y=-x) = 1.