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

1, 1, 1, 3, 3, 15, 10, 70, 35, 315, 126, 1386, 462, 6006, 1716, 25740, 6435, 109395, 24310, 461890, 92378, 1939938, 352716, 8112468, 1352078, 33801950, 5200300, 140408100, 20058300, 581690700, 77558760, 2404321560, 300540195, 9917826435, 1166803110, 40838108850

Offset: 0

Author

Peter Luschny, Aug 15 2016

Keywords

• nonn

Crossrefs

Cf. A001700, A033876, A088218, A056040.

Programs

Maple

st := (x*(1-4*x^2)^(-3/2)+(1-4*x^2)^(-1/2)+x+1)/2: series(st,x,36):
PolynomialTools:-CoefficientList(convert(%,polynom),x);

Mathematica

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

Sage

def A275324():
    r, n = 2, 1
    yield 1
    yield 1
    while True:
        n += 1
        r *= 4/n if is_even(n) else n
        yield r // 4
a = A275324(); print([next(a) for i in range(16)])

Formula

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

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

For a recurrence see the Sage script.

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

From Amiram Eldar, Mar 04 2023: (Start)

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

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