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A275324 Expansion of (x*(1-4*x^2)^(-3/2) + (1-4*x^2)^(-1/2) + x + 1)/2.

%I A275324 #22 Mar 04 2023 05:00:44
%S A275324 1,1,1,3,3,15,10,70,35,315,126,1386,462,6006,1716,25740,6435,109395,
%T A275324 24310,461890,92378,1939938,352716,8112468,1352078,33801950,5200300,
%U A275324 140408100,20058300,581690700,77558760,2404321560,300540195,9917826435,1166803110,40838108850
%N A275324 Expansion of (x*(1-4*x^2)^(-3/2) + (1-4*x^2)^(-1/2) + x + 1)/2.
%F A275324 Interweaved from (1+(1-4*x)^(-1/2))/2 (compare A088218 & A001700) and (1+(1-4*x)^(-3/2))/2 (compare A033876).
%F A275324 E.g.f.: (1 + x)*(1 + BesselI(0, 2*x))/2.
%F A275324 For a recurrence see the Sage script.
%F A275324 a(n) = A056040(n)/2 for n>=2.
%F A275324 From _Amiram Eldar_, Mar 04 2023: (Start)
%F A275324 Sum_{n>=0} 1/a(n) = 2/3 + 16*Pi/(9*sqrt(3)).
%F A275324 Sum_{n>=0} (-1)^n/a(n) = 8/3 - 8*Pi/(9*sqrt(3)). (End)
%p A275324 st := (x*(1-4*x^2)^(-3/2)+(1-4*x^2)^(-1/2)+x+1)/2: series(st,x,36):
%p A275324 PolynomialTools:-CoefficientList(convert(%,polynom),x);
%t A275324 Table[If[n<2,1,n!/Quotient[n,2]!^2/2], {n,0,30}]
%t A275324 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 A275324 (Sage)
%o A275324 def A275324():
%o A275324     r, n = 2, 1
%o A275324     yield 1
%o A275324     yield 1
%o A275324     while True:
%o A275324         n += 1
%o A275324         r *= 4/n if is_even(n) else n
%o A275324         yield r // 4
%o A275324 a = A275324(); print([next(a) for i in range(16)])
%Y A275324 Cf. A001700, A033876, A088218, A056040.
%K A275324 nonn
%O A275324 0,4
%A A275324 _Peter Luschny_, Aug 15 2016