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%I #24 Jan 09 2019 16:07:33
%S 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,
%T 256,3281,12352,23801,29056,23801,12352,3281,256,1,1,1024,29525,
%U 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
%N 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.
%C See A322621 for another description of the e.g.f. of this sequence.
%H Paul D. Hanna, <a href="/A322193/b322193.txt">Table of n, a(n) for n = 0..2600 terms of this triangle as read by rows 0..50.</a>
%F E.g.f.: C(x,y) and related series S(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 C(x, y=0) = cosh(x).
%F C(x, y=x) = cosh(x)^2 / (1 - sinh(x)^2).
%F C(x, y=-x) = 1.
%e 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!) + ...
%e where C(x,y) = cosh(x)*cosh(y) / (1 - sinh(x)*sinh(y)).
%e This irregular triangle of coefficients of x^(2*n-k)*y^k/((2*n-k)!*k!) in C(x,y) begins
%e 1;
%e 1, 1, 1;
%e 1, 4, 5, 4, 1;
%e 1, 16, 41, 52, 41, 16, 1;
%e 1, 64, 365, 784, 977, 784, 365, 64, 1;
%e 1, 256, 3281, 12352, 23801, 29056, 23801, 12352, 3281, 256, 1;
%e 1, 1024, 29525, 196864, 589217, 1049344, 1257125, 1049344, 589217, 196864, 29525, 1024, 1;
%e 1, 4096, 265721, 3146752, 14677961, 37789696, 63318641, 74628352, 63318641; 37789696, 14677961, 3146752, 265721, 4096, 1; ...
%e RELATED SERIES.
%e The series S(x,y), such that C(x,y)^2 - S(x,y)^2 = 1, begins
%e 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 The e.g.f. may be written with coefficients of x^(2*n-k)*y^k/(2*n)!, as follows:
%e 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! + ...
%e these coefficients are described by triangle A322621.
%t T[n_, k_] := (2n-k)! k! SeriesCoefficient[Cosh[x] Cosh[y]/(1-Sinh[x] Sinh[y]), {x, 0, 2n-k}, {y, 0, k}];
%t Table[T[n, k], {n, 0, 8}, {k, 0, 2n}] // Flatten (* _Jean-François Alcover_, Dec 29 2018 *)
%o (PARI) {T(n, k) = my(X=x+x*O(x^(2*n-k)), Y=y+y*O(y^k));
%o C = cosh(X)*cosh(Y)/(1 - sinh(X)*sinh(Y));
%o (2*n-k)!*k!*polcoeff(polcoeff(C, 2*n-k, x), k, y)}
%o /* Print as a triangle */
%o for(n=0, 10, for(k=0, 2*n, print1( T(n, k), ", ")); print(""))
%Y Cf. A322190 (C + S), A322194 (S), A322195 (main diagonal).
%K nonn,tabf
%O 0,6
%A _Paul D. Hanna_, Dec 20 2018
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