A322194 E.g.f.: S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)), where S(x,y) = Sum_{n>=0} Sum_{k=0..2*n+1} T(n,k) * x^(2*n+1-k)*y^k/((2*n+1-k)!*k!), as a triangle of coefficients T(n,k) read by rows.

1, 1, 1, 2, 2, 1, 1, 8, 14, 14, 8, 1, 1, 32, 122, 200, 200, 122, 32, 1, 1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1, 1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1, 1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1, 1, 8192, 797162, 12584960, 73330760, 226744832, 446442272, 614111360, 614111360, 446442272, 226744832, 73330760, 12584960, 797162, 8192, 1

Offset

1,4

Comments

See A322622 for another description of the e.g.f. of this sequence.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..2652

Formula

E.g.f.: S(x,y) and related series C(x,y) satisfy the following identities.

(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).

(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).

(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)).

(6c) C(x,y) + S(x,y) = (cosh(x) + sinh(x)*cosh(y)) / (cosh(y) - sinh(y)*cosh(x)).

(6d) C(x,y) + S(x,y) = (cosh(y) + sinh(y)*cosh(x)) / (cosh(x) - sinh(x)*cosh(y)).

SPECIAL ARGUMENTS.

S(x, y=0) = sinh(x).

S(x, y=x) = 2*sinh(x) / (1 - sinh(x)^2).

S(x, y=-x) = 0.

Example

E.g.f.: 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 S(x,y) = (sinh(x) + sinh(y)) / (1 - sinh(x)*sinh(y)).
This irregular triangle of coefficients of x^(2*n+1-k)*y^k/((2*n+1-k)!*k!) in C(x,y) begins
1, 1;
1, 2, 2, 1;
1, 8, 14, 14, 8, 1;
1, 32, 122, 200, 200, 122, 32, 1;
1, 128, 1094, 3104, 4808, 4808, 3104, 1094, 128, 1;
1, 512, 9842, 49280, 118280, 174752, 174752, 118280, 49280, 9842, 512, 1;
1, 2048, 88574, 786944, 2939528, 6297728, 8948384, 8948384, 6297728, 2939528, 786944, 88574, 2048, 1;
1, 8192, 797162, 12584960, 73330760, 226744832, 446442272, 614111360, 614111360, 446442272, 226744832, 73330760, 12584960, 797162, 8192, 1; ...
RELATED SERIES.
The series C(x,y), such that C(x,y)^2 - S(x,y)^2 = 1, begins
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!) + ...
The e.g.f. may be written with coefficients of x^(2*n+1-k)*y^k/(2*n+1)!, as follows:
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! + ...
these coefficients are described by triangle A322622.

Mathematica

T[n_, k_] := (2n-k+1)! k! SeriesCoefficient[(Sinh[x] + Sinh[y])/(1 - Sinh[x] Sinh[y]), {x, 0, 2n-k+1}, {y, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, 2n+1}] // Flatten (* Jean-François Alcover, Dec 29 2018 *)

Prog

(PARI) {T(n, k) = my(X=x+x*O(x^(2*n+1-k)), Y=y+y*O(y^k));
S = (sinh(X) + sinh(Y))/(1 - sinh(X)*sinh(Y));
(2*n+1-k)!*k!*polcoeff(polcoeff(S, 2*n+1-k, x), k, y)}
/* Print as a triangle */
for(n=0, 10, for(k=0, 2*n+1, print1( T(n, k), ", ")); print(""))

Crossrefs

Cf. A322190 (C + S), A322193 (C), A322196 (main diagonal).

Cf. A322622.

Sequence in context: A260360 A011296 A176602 * A174120 A240939 A016739

Adjacent sequences: A322191 A322192 A322193 * A322195 A322196 A322197

Keyword

nonn,tabf

Author

Paul D. Hanna, Dec 20 2018

Status

approved