A285046 Expansion of cosh(9*arctanh(2*sqrt(x))).

1, 162, 4806, 71892, 758214, 6506172, 48783900, 332715240, 2115552582, 12745645484, 73577414196, 410265444888, 2222886926364, 11756568121560, 60911288332920, 310024235290320, 1553692427724870

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, A285045.

Programs

Maple

seq(1/105*(4096*n^4 + 512*n^3 + 3392*n^2 + 400*n + 105)*binomial(2*n,n), n = 0..20);

PARI

x='x + O('x^30); print(Vec((1 + 144*x + 2016*x^2 + 5376*x^3 + 2304*x^4)/(1 - 4*x)^(9/2))) \\ Indranil Ghosh, Apr 10 2017

Formula

a(n) = 1/105*(4096*n^4 + 512*n^3 + 3392*n^2 + 400*n + 105)*binomial(2*n,n).

O.g.f. cosh(9*arctanh(2*sqrt(x))) = (1 + 144*x + 2016*x^2 + 5376*x^3 + 2304x^4)/(1 - 4*x)^(9/2) = 1 + 162*x + 4806*x^2 + 71892*x^3 + ....

Note that the zeros of the polynomial 1 + 144*x^2 + 2016*x^4 + 5376*x^6 + 2304*x^8 = 1/2*((1 + 2*x)^9 + (1 - 2*x)^9), are given by 1/2*cot(k*Pi/9)*i for 1 <= k <= 8. See A085840.

O.g.f. for the sequence with interpolated zeros: 1/2*( ((1 + 2*x)/(1 - 2*x))^(9/2) + ((1 - 2*x)/(1 + 2*x))^(9/2) ) = 1 + 162*x^2 + 4806*x^4 + 71892*x^6 + ....