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A322620
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, 2, 1, 1, 6, 6, 1, 1, 16, 30, 16, 1, 1, 40, 140, 140, 40, 1, 1, 96, 615, 1040, 615, 96, 1, 1, 224, 2562, 7000, 7000, 2562, 224, 1, 1, 512, 10220, 43904, 68390, 43904, 10220, 512, 1, 1, 1152, 39384, 260736, 605808, 605808, 260736, 39384, 1152, 1, 1, 2560, 147645, 1482240, 4998210, 7322112, 4998210, 1482240, 147645, 2560, 1, 1, 5632, 541310, 8131200, 39032400, 80735424, 80735424, 39032400, 8131200, 541310, 5632, 1, 1, 12288, 1948650, 43310080, 291662415, 831080448, 1161583500, 831080448, 291662415, 43310080, 1948650, 12288, 1
OFFSET
0,5
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 A322190 for another description of the e.g.f. of this sequence.
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) = binomial(n,k) * A322190(n,k).
Sum_{k=0..n} 2^k * T(n,k) = A245140(n).
Sum_{k=0..n} 3^k * T(n,k) = A245155(n).
Sum_{k=0..n} 2^(n-k) * 3^k * T(n,k) = A245166(n).
EXAMPLE
E.g.f.: 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! + ...
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, 2, 6, 16, 40, 96, 224, 512, 1152, ...;
1, 6, 30, 140, 615, 2562, 10220, 39384, 147645, ...;
1, 16, 140, 1040, 7000, 43904, 260736, 1482240, 8131200, ...;
1, 40, 615, 7000, 68390, 605808, 4998210, 39032400, 291662415, ...;
1, 96, 2562, 43904, 605808, 7322112, 80735424, 831080448, 8105175936, ...;
1, 224, 10220, 260736, 4998210, 80735424, 1161583500, 15355426944, ...;
1, 512, 39384, 1482240, 39032400, 831080448, 15355426944, 256124504064, ...; ...
This sequence may be written as a triangle, starting as
1,
1, 1,
1, 2, 1,
1, 6, 6, 1;
1, 16, 30, 16, 1;
1, 40, 140, 140, 40, 1;
1, 96, 615, 1040, 615, 96, 1;
1, 224, 2562, 7000, 7000, 2562, 224, 1;
1, 512, 10220, 43904, 68390, 43904, 10220, 512, 1;
1, 1152, 39384, 260736, 605808, 605808, 260736, 39384, 1152, 1;
1, 2560, 147645, 1482240, 4998210, 7322112, 4998210, 1482240, 147645, 2560, 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*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! + ...
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! + ...
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! + 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!) + ...
these coefficients are described by table A322190.
MATHEMATICA
nmax = 12;
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)!;
tt = Table[t[n, k], {n, 0, nmax}, {k, 0, nmax}];
T[n_, k_] := tt[[n+1, k+1]];
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. A322621 (C(x,y)), A322622 (S(x,y)), A322623 (antidiagonal sums), A322624 (main diagonal), A322625, A057711 (column 1).
Sequence in context: A329228 A172373 A174411 * A376935 A155795 A009963
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 20 2018
STATUS
approved