A285045 Expansion of cosh(7*arctanh(2*sqrt(x))).

1, 98, 1862, 19796, 160454, 1114428, 7008540, 41132520, 229435206, 1230873644, 6403088692, 32488200472, 161473267228, 788758622680, 3796375603320, 18040943163600, 84786596572230, 394599588033420, 1820669979129540, 8335975464699960

Offset: 0

Author

Peter Bala, Apr 10 2017

Keywords

• nonn
• easy

Comments

Note that the function cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. See, for example, A010006.

In A285043 through A285046 we consider sequences with o.g.f. cosh((2*n+1)*arctanh(2*sqrt(x))) for n = 1, 2, 3 and 4: n = 0 gives the central binomial coefficients A000984.

Crossrefs

Cf. A000984, A010006, A085840, A285043, A285044, A285046.

Programs

Maple

seq(1/15*(512*n^3 + 64*n^2 + 144*n + 15)*binomial(2*n,n), n = 0..20);

Mathematica

CoefficientList[Series[Cosh[7*ArcTanh[2Sqrt[x]]],{x,0,20}],x] (* Harvey P. Dale, Jun 07 2024 *)

Formula

a(n) = 1/15*(512*n^3 + 64*n^2 + 144*n + 15)*binomial(2*n,n).

O.g.f. cosh(7*arctanh(2*sqrt(x))) = (1 + 84*x + 560*x^2 + 448*x^3)/(1 - 4*x)^(7/2) = 1 + 98*x + 1862*x^2 + 19796*x^3 + ....

Note that the zeros of the polynomial 1 + 84*x^2 + 560*x^4 + 448*x^6 = 1/2*((1 + 2*x)^7 + (1 - 2*x)^7), are given by 1/2*cot(k*Pi/7)*i for 1 <= k <= 6. See A085840.

O.g.f. for the sequence with interpolated zeros: 1/2*( ((1 + 2*x)/(1 - 2*x))^(7/2) + ((1 - 2*x)/(1 + 2*x))^(7/2) ) = 1 + 98*x^2 + 1862*x^4 + 19796*x^6 + ....

D-finite with recurrence: n*(2*n-1)*a(n) +2*(-8*n^2+16*n-57)*a(n-1) +16*(2*n-3)*(n-2)*a(n-2)=0. - R. J. Mathar, Jan 22 2020