A275324 - OEIS

%I #22 Mar 04 2023 05:00:44

%S 1,1,1,3,3,15,10,70,35,315,126,1386,462,6006,1716,25740,6435,109395,

%T 24310,461890,92378,1939938,352716,8112468,1352078,33801950,5200300,

%U 140408100,20058300,581690700,77558760,2404321560,300540195,9917826435,1166803110,40838108850

%N Expansion of (x*(1-4*x^2)^(-3/2) + (1-4*x^2)^(-1/2) + x + 1)/2.

%F Interweaved from (1+(1-4*x)^(-1/2))/2 (compare A088218 & A001700) and (1+(1-4*x)^(-3/2))/2 (compare A033876).

%F E.g.f.: (1 + x)*(1 + BesselI(0, 2*x))/2.

%F For a recurrence see the Sage script.

%F a(n) = A056040(n)/2 for n>=2.

%F From _Amiram Eldar_, Mar 04 2023: (Start)

%F Sum_{n>=0} 1/a(n) = 2/3 + 16*Pi/(9*sqrt(3)).

%F Sum_{n>=0} (-1)^n/a(n) = 8/3 - 8*Pi/(9*sqrt(3)). (End)

%p st := (x*(1-4*x^2)^(-3/2)+(1-4*x^2)^(-1/2)+x+1)/2: series(st,x,36):

%p PolynomialTools:-CoefficientList(convert(%,polynom),x);

%t Table[If[n<2,1,n!/Quotient[n,2]!^2/2], {n,0,30}]

%t CoefficientList[Series[(x*(1 - 4*x^2)^(-3/2) + (1 - 4*x^2)^(-1/2) + x + 1)/2, {x, 0, 50}], x] (* _G. C. Greubel_, Aug 15 2016 *)

%o (Sage)

%o def A275324():

%o     r, n = 2, 1

%o     yield 1

%o     yield 1

%o     while True:

%o         n += 1

%o         r *= 4/n if is_even(n) else n

%o         yield r // 4

%o a = A275324(); print([next(a) for i in range(16)])

%Y Cf. A001700, A033876, A088218, A056040.

%K nonn

%O 0,4

%A _Peter Luschny_, Aug 15 2016