A285045 - OEIS

%I #18 Nov 22 2024 00:35:11

%S 1,98,1862,19796,160454,1114428,7008540,41132520,229435206,1230873644,

%T 6403088692,32488200472,161473267228,788758622680,3796375603320,

%U 18040943163600,84786596572230,394599588033420,1820669979129540,8335975464699960

%N Expansion of cosh(7*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) = 1/15*(512*n^3 + 64*n^2 + 144*n + 15)*binomial(2*n,n).

%F 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 + ....

%F 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.

%F 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 + ....

%F 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

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

%t CoefficientList[Series[Cosh[7*ArcTanh[2Sqrt[x]]],{x,0,20}],x] (* _Harvey P. Dale_, Jun 07 2024 *)

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

%K nonn,easy

%O 0,2

%A _Peter Bala_, Apr 10 2017