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%I #55 Dec 29 2018 10:35:11
%S 1,1,1,1,1,1,1,2,2,1,1,4,5,4,1,1,8,14,14,8,1,1,16,41,52,41,16,1,1,32,
%T 122,200,200,122,32,1,1,64,365,784,977,784,365,64,1,1,128,1094,3104,
%U 4808,4808,3104,1094,128,1,1,256,3281,12352,23801,29056,23801,12352,3281,256,1,1,512,9842,49280,118280,174752,174752,118280,49280,9842,512,1,1,1024,29525,196864,589217,1049344,1257125,1049344,589217,196864,29525,1024,1,1,2048,88574,786944,2939528,6297728,8948384,8948384,6297728,2939528,786944,88574,2048,1
%N E.g.f.: A(x,y) = (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), where A(x,y) = Sum_{n>=0} Sum_{k>=0} T(n,k) * x^n*y^k/(n!*k!), as a square table of coefficients T(n,k) read by antidiagonals.
%C Compare to the addition theorem of Jacobi's elliptic functions: cn(x+y) + i*sn(x+y) = (cn(x) + i*sn(x)*dn(y)) * (cn(y) + i*sn(y)*dn(x)) / (1 - k^2*sn(x)^2*sn(y)^2), where the modulus k is implicit.
%C See A322620 for another description of the e.g.f. of this sequence.
%H Paul D. Hanna, <a href="/A322190/b322190.txt">Table of n, a(n) for n = 0..1325 terms of this square table read by antidiagonals across rows 0..50.</a>
%F E.g.f.: A(x,y) = (cosh(x) + sinh(x)*cosh(y)) * (cosh(y) + sinh(y)*cosh(x)) / (1 - sinh(x)^2*sinh(y)^2).
%F E.g.f.: A(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)).
%F E.g.f.: A(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)).
%F E.g.f.: A(x,y) = C(x,y) + S(x,y) such that the following identities hold.
%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 (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 SPECIAL ARGUMENTS.
%F A(x, y=0) = exp(x).
%F A(x, y=x) = (1 + sinh(x)) / (1 - sinh(x)).
%F A(x, y=-x) = 1.
%F FORMULAS FOR TERMS.
%F a(n) = A322620(n,k) / binomial(n,k).
%F Sum_{k=0..n} 2^k * binomial(n,k) * T(n,k) = A245140(n).
%F Sum_{k=0..n} 3^k * binomial(n,k) * T(n,k) = A245155(n).
%F Sum_{k=0..n} 2^(n-k) * 3^k * binomial(n,k) * T(n,k) = A245166(n).
%e E.g.f.: A(x,y) = 1 + (1*x + 1*y) + (1*x^2/2! + 1*x*y + 1*y^2/2!) + (1*x^3/3! + 2*x^2*y/2! + 2*x*y^2/2! + 1*y^3/3!) + (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^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^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^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!) + (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 A(x,y) = (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
%e This square table of coefficients of x^n*y^k/(n!*k!) in A(x,y) begins
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
%e 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...;
%e 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, ...;
%e 1, 4, 14, 52, 200, 784, 3104, 12352, 49280, 196864, ...;
%e 1, 8, 41, 200, 977, 4808, 23801, 118280, 589217, 2939528, ...;
%e 1, 16, 122, 784, 4808, 29056, 174752, 1049344, 6297728, 37789696, ...;
%e 1, 32, 365, 3104, 23801, 174752, 1257125, 8948384, 63318641, 446442272, ...;
%e 1, 64, 1094, 12352, 118280, 1049344, 8948384, 74628352, 614111360, 5010663424, ...;
%e 1, 128, 3281, 49280, 589217, 6297728, 63318641, 614111360, 5823720257, 54420050048, ...; ...
%e This sequence may be written as a triangle, starting as
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 2, 1;
%e 1, 4, 5, 4, 1;
%e 1, 8, 14, 14, 8, 1;
%e 1, 16, 41, 52, 41, 16, 1;
%e 1, 32, 122, 200, 200, 122, 32, 1;
%e 1, 64, 365, 784, 977, 784, 365, 64, 1;
%e 1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1;
%e 1, 256, 3281, 12352, 23801, 29056, 23801, 12352, 3281, 256, 1;
%e 1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1; ...
%e RELATED SERIES.
%e The series expansions for C(x,y) and S(x,y) are given by
%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 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 A(x,y) = C(x,y) + S(x,y) such that C(x,y)^2 - S(x,y)^2 = 1.
%e The e.g.f. may be written with coefficients of x^n*y^k/(n+k)!, as follows:
%e A(x,y) = 1 + (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2)/2! + (1*x^3 + 6*x^2*y + 6*x*y^2 + 1*y^3)/3! + (1*x^4 + 16*x^3*y + 30*x^2*y^2 + 16*x*y^3 + 1*y^4)/4! + (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^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^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! + (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 table A322620.
%t nmax = 13;
%t t[n_, k_] := SeriesCoefficient[(Cosh[x] Cosh[y] + Sinh[x] + Sinh[y])/(1 - Sinh[x] Sinh[y]), {x, 0, n}, {y, 0, k}] n! k!;
%t Table[t[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 26 2018 *)
%o (PARI) {T(n,k) = my(X=x+x*O(x^n),Y=y+y*O(y^k));
%o C = cosh(X)*cosh(Y)/(1 - sinh(X)*sinh(Y));
%o S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));
%o n!*k!*polcoeff(polcoeff(C + S,n,x),k,y)}
%o /* Print as a square table */
%o for(n=0,10,for(k=0,10,print1( T(n,k),", "));print(""))
%o /* Print as a triangle */
%o for(n=0,15,for(k=0,n,print1( T(n-k,k),", "));print(""))
%Y Cf. A322193 (C(x,y)), A322194 (S(x,y)), A322195 (main diagonal), A322196, A322197.
%Y Cf. A322620, A245140, A245155, A245166.
%K nonn,tabl
%O 0,8
%A _Paul D. Hanna_, Dec 19 2018
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