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A256093
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G.f.: 2 - x*2/(1 - (1-8*x)^(1/4)).
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1
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1, 3, 5, 20, 101, 572, 3470, 22040, 144669, 973356, 6676186, 46503080, 328034226, 2338460056, 16819478972, 121903180848, 889396747869, 6526715628492, 48141140144546, 356708675726088, 2653863473928870, 19816831149068360, 148466651633265540, 1115659552758534480
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{k=0..(n+1)} binomial(2*k-2,k)*2^(n-k+1)*binomial(2*n-k,n-k+1))/n, a(0)=1.
a(n) ~ 2^(3*n-4) / (Gamma(3/4) * n^(5/4)) * (1 + 2*Gamma(3/4) / (n^(1/4) * sqrt(Pi)) + 3*Gamma(3/4)^2 / (Pi*sqrt(2*n))). - Vaclav Kotesovec, Mar 15 2015
(512*n^3 - 768*n^2 + 352*n - 48)*a(n) + (-192*n^3 - 192*n^2 - 12*n - 12)*a(n + 1) + (24*n^3 + 84*n^2 + 84*n + 24)*a(n + 2) + (-n^3 - 6*n^2 - 11*n - 6)*a(n + 3) = 0 for n >= 1. - Robert Israel, Jan 20 2020
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MAPLE
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f:= gfun:-rectoproc({(512*n^3 - 768*n^2 + 352*n - 48)*a(n) + (-192*n^3 - 192*n^2 - 12*n - 12)*a(n + 1) + (24*n^3 + 84*n^2 + 84*n + 24)*a(n + 2) + (-n^3 - 6*n^2 - 11*n - 6)*a(n + 3), a(0) = 1, a(1) = 3, a(2) = 5, a(3) = 20}, a(n), remember):
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MATHEMATICA
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CoefficientList[Series[2-x*2/(1-(1-8*x)^(1/4)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 15 2015 *)
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PROG
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(Maxima)
a(n):=if n=0 then 1 else sum(binomial(2*k-2, k)*2^(n-k+1)*binomial(2*n-k, n-k+1), k, 0, n+1)/n;
(PARI) x='x+O('x^50); Vec(2-x*2/(1-(1-8*x)^(1/4))) \\ G. C. Greubel, Jun 03 2017
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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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