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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

Paul D. Hanna, Table of n, a(n) for n = 0..1325 terms of this square table read by antidiagonals across rows 0..50.

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-Fran├žois 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 May 12 17:19 EDT 2021. Contains 343829 sequences. (Running on oeis4.)