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A344504
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a(n) = [x^n] ((x - 1)/sqrt(4/(x + 1) - 3) + x + 1)/(2*x*(3*x - 1)).
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0
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0, 1, 6, 26, 100, 361, 1254, 4245, 14108, 46247, 149998, 482412, 1540880, 4893859, 15468910, 48696930, 152764452, 477771447, 1490245302, 4637349186, 14400224496, 44632551567, 138101593398, 426658380621, 1316306945952, 4055853282741, 12482506508174, 38375733088400
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OFFSET
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0,3
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COMMENTS
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Motzkin transform of the squares.
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} k^2*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).
a(n) ~ 4 * 3^(n - 1/2) * sqrt(n/Pi) * (1 - sqrt(3*Pi/n)/2). - Vaclav Kotesovec, May 24 2021
D-finite with recurrence -(n+1)*(2*n-3)*a(n) +(10*n^2-5*n-12)*a(n-1) -3*(2*n+5)*(n-1)*a(n-2) -9*(2*n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Mar 06 2022
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MAPLE
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gf := ((x - 1)/sqrt(4/(x + 1) - 3) + x + 1)/(2*x*(3*x - 1)):
ser := series(gf, x, 30): seq(coeff(ser, x, n), n=0..27);
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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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