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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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

LINKS

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

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.

EXAMPLE

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.

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-Fran├žois Alcover, Dec 29 2018 *)

PROG

(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(""))

CROSSREFS

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

Sequence in context: A290558 A071992 A291845 * A174984 A092141 A120867

Adjacent sequences:  A322190 A322191 A322192 * A322194 A322195 A322196

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Dec 20 2018

STATUS

approved

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Last modified November 14 20:15 EST 2019. Contains 329130 sequences. (Running on oeis4.)