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A112556
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Sums of squared terms in rows of triangle A112555.
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4
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1, 2, 2, 4, 10, 32, 112, 408, 1514, 5680, 21472, 81644, 311896, 1196132, 4602236, 17757184, 68680170, 266200112, 1033703056, 4020716124, 15662273840, 61092127492, 238582873476, 932758045124, 3650336341240, 14298633670932, 56055986383412, 219931273282348, 863504076182884, 3392593262288780, 13337336618626132
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OFFSET
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0,2
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COMMENTS
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First differences form A072547 and equals the unsigned central terms of triangle A112555.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
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FORMULA
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G.f.: ( 2*(1+x)/(1-x) + x/(1-4*x)^(1/2) )/(2+x).
a(n) ~ 2^(2*n) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 26 2016
a(n) = (1/3)*(4 - (-1/2)^n) - Sum_{j=0..n-1} binomial(2*j, j)*(-1/2)^(n-j). - G. C. Greubel, Jan 13 2022
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MATHEMATICA
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CoefficientList[Series[(2(1+x)/(1-x)+x/(1-4x)^(1/2))/(2+x), {x, 0, 30}], x] (* Harvey P. Dale, May 26 2011 *)
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PROG
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(PARI) {a(n)=local(x=X+X*O(X^n)); polcoeff((2*(1+x)/(1-x)+x/(1-4*x)^(1/2))/(2+x), n, X)}
(Magma) [(1/3)*(4 - (-1/2)^n) + (n+1)*Catalan(n) - (&+[(j+1)*Catalan(j)*(-1/2)^(n-j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 13 2022
(Sage) [(1/3)*(4 - (-1/2)^n) - sum( binomial(2*j, j)*(-1/2)^(n-j) for j in (0..n-1)) for n in (0..30)] # G. C. Greubel, Jan 13 2022
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CROSSREFS
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Cf. A072547, A112555.
Sequence in context: A309159 A002420 A284016 * A254400 A054100 A034165
Adjacent sequences: A112553 A112554 A112555 * A112557 A112558 A112559
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Sep 21 2005
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STATUS
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approved
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