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 A322190 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. 10
 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, 122, 200, 200, 122, 32, 1, 1, 64, 365, 784, 977, 784, 365, 64, 1, 1, 128, 1094, 3104, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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. See A322620 for another description of the e.g.f. of this sequence. LINKS FORMULA 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). E.g.f.: A(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)). E.g.f.: A(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)). E.g.f.: A(x,y) = C(x,y) + S(x,y) such that the following identities hold. (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). (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)). SPECIAL ARGUMENTS. A(x, y=0) = exp(x). A(x, y=x) = (1 + sinh(x)) / (1 - sinh(x)). A(x, y=-x) = 1. FORMULAS FOR TERMS. a(n) = A322620(n,k) / binomial(n,k). Sum_{k=0..n} 2^k * binomial(n,k) * T(n,k) = A245140(n). Sum_{k=0..n} 3^k * binomial(n,k) * T(n,k) = A245155(n). Sum_{k=0..n} 2^(n-k) * 3^k * binomial(n,k) * T(n,k) = A245166(n). EXAMPLE 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!) + ... where A(x,y) = (cosh(x)*cosh(y) + sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)). This square table of coefficients of x^n*y^k/(n!*k!) in A(x,y) begins 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...; 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...; 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, ...; 1, 4, 14, 52, 200, 784, 3104, 12352, 49280, 196864, ...; 1, 8, 41, 200, 977, 4808, 23801, 118280, 589217, 2939528, ...; 1, 16, 122, 784, 4808, 29056, 174752, 1049344, 6297728, 37789696, ...; 1, 32, 365, 3104, 23801, 174752, 1257125, 8948384, 63318641, 446442272, ...; 1, 64, 1094, 12352, 118280, 1049344, 8948384, 74628352, 614111360, 5010663424, ...; 1, 128, 3281, 49280, 589217, 6297728, 63318641, 614111360, 5823720257, 54420050048, ...; ... This sequence may be written as a triangle, starting as 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, 122, 200, 200, 122, 32, 1; 1, 64, 365, 784, 977, 784, 365, 64, 1; 1, 128, 1094, 3104, 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; ... RELATED SERIES. The series expansions for C(x,y) and S(x,y) are given by 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!) + ... 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!) + ... where A(x,y) = C(x,y) + S(x,y) such that C(x,y)^2 - S(x,y)^2 = 1. The e.g.f. may be written with coefficients of x^n*y^k/(n+k)!, as follows: 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! + ... these coefficients are described by table A322620. MATHEMATICA nmax = 13; 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!; Table[t[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 26 2018 *) PROG (PARI) {T(n, k) = my(X=x+x*O(x^n), Y=y+y*O(y^k)); C = cosh(X)*cosh(Y)/(1 - sinh(X)*sinh(Y)); S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y)); n!*k!*polcoeff(polcoeff(C + S, n, x), k, y)} /* Print as a square table */ for(n=0, 10, for(k=0, 10, print1( T(n, k), ", ")); print("")) /* Print as a triangle */ for(n=0, 15, for(k=0, n, print1( T(n-k, k), ", ")); print("")) CROSSREFS Cf. A322193 (C(x,y)), A322194 (S(x,y)), A322195 (main diagonal), A322196, A322197. Cf. A322620, A245140, A245155, A245166. Sequence in context: A292189 A284992 A191687 * A177254 A340910 A132311 Adjacent sequences: A322187 A322188 A322189 * A322191 A322192 A322193 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Dec 19 2018 STATUS approved

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Last modified January 29 11:27 EST 2023. Contains 359922 sequences. (Running on oeis4.)