A285046 - OEIS

%I #13 Nov 22 2024 00:35:07

%S 1,162,4806,71892,758214,6506172,48783900,332715240,2115552582,

%T 12745645484,73577414196,410265444888,2222886926364,11756568121560,

%U 60911288332920,310024235290320,1553692427724870

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

%C 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.

%C 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.

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

%F 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 + ....

%F 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.

%F 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 + ....

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

%o (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

%Y Cf. A000984, A010006, A085840, A285043, A285044, A285045.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Apr 10 2017