%I #24 Mar 04 2020 16:26:20

%S 1,1,1,2,2,1,1,8,14,14,8,1,1,32,122,200,200,122,32,1,1,128,1094,3104,

%T 4808,4808,3104,1094,128,1,1,512,9842,49280,118280,174752,174752,

%U 118280,49280,9842,512,1,1,2048,88574,786944,2939528,6297728,8948384,8948384,6297728,2939528,786944,88574,2048,1,1,8192,797162,12584960,73330760,226744832,446442272,614111360,614111360,446442272,226744832,73330760,12584960,797162,8192,1

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

%C See A322622 for another description of the e.g.f. of this sequence.

%H Paul D. Hanna, <a href="/A322194/b322194.txt">Table of n, a(n) for n = 1..2652</a>

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

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

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

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

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

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

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

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

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

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

%F (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).

%F (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).

%F (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).

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

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

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

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

%F SPECIAL ARGUMENTS.

%F S(x, y=0) = sinh(x).

%F S(x, y=x) = 2*sinh(x) / (1 - sinh(x)^2).

%F S(x, y=-x) = 0.

%e E.g.f.: 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!) + ...

%e where S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).

%e This irregular triangle of coefficients of x^(2*n+1-k)*y^k/((2*n+1-k)!*k!) in C(x,y) begins

%e 1, 1;

%e 1, 2, 2, 1;

%e 1, 8, 14, 14, 8, 1;

%e 1, 32, 122, 200, 200, 122, 32, 1;

%e 1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1;

%e 1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1;

%e 1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1;

%e 1, 8192, 797162, 12584960, 73330760, 226744832, 446442272, 614111360, 614111360, 446442272, 226744832, 73330760, 12584960, 797162, 8192, 1; ...

%e RELATED SERIES.

%e The series C(x,y), such that C(x,y)^2 - S(x,y)^2 = 1, begins

%e 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!) + ...

%e The e.g.f. may be written with coefficients of x^(2*n+1-k)*y^k/(2*n+1)!, as follows:

%e S(x,y) = (1*x + 1*y) + (1*x^3 + 6*x^2*y + 6*x*y^2 + 1*y^3)/3! + (1*x^5 + 40*x^4*y + 140*x^3*y^2 + 140*x^2*y^3 + 40*x*y^4 + 1*y^5)/5! + (1*x^7 + 224*x^6*y + 2562*x^5*y^2 + 7000*x^4*y^3 + 7000*x^3*y^4 + 2562*x^2*y^5 + 224*x*y^6 + 1*y^7)/7! + ...

%e these coefficients are described by triangle A322622.

%t T[n_, k_] := (2n-k+1)! k! SeriesCoefficient[(Sinh[x] + Sinh[y])/(1 - Sinh[x] Sinh[y]), {x, 0, 2n-k+1}, {y, 0, k}];

%t Table[T[n, k], {n, 0, 7}, {k, 0, 2n+1}] // Flatten (* _Jean-François Alcover_, Dec 29 2018 *)

%o (PARI) {T(n, k) = my(X=x+x*O(x^(2*n+1-k)), Y=y+y*O(y^k));

%o S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));

%o (2*n+1-k)!*k!*polcoeff(polcoeff(S, 2*n+1-k, x), k, y)}

%o /* Print as a triangle */

%o for(n=0, 10, for(k=0, 2*n+1, print1( T(n, k), ", ")); print(""))

%Y Cf. A322190 (C + S), A322193 (C), A322196 (main diagonal).

%Y Cf. A322622.

%K nonn,tabf

%O 1,4

%A _Paul D. Hanna_, Dec 20 2018