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A275324
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Expansion of (x*(1-4*x^2)^(-3/2) + (1-4*x^2)^(-1/2) + x + 1)/2.
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2
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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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.
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)
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(Sage)
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)])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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