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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 29 18:13 EST 2022. Contains 358431 sequences. (Running on oeis4.)