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A254408
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a(n) = 2*n^2*binomial(2*n,n)^2, a closed form for a double binomial sum involving absolute values.
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5
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0, 8, 288, 7200, 156800, 3175200, 61471872, 1154305152, 21201523200, 382952512800, 6826955907200, 120427502203008, 2105988385632768, 36562298361680000, 630861905459520000, 10827650254927680000, 184984389244186675200
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OFFSET
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0,2
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..500
Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, and Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477 [math.CO], 2014.
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FORMULA
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a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k^2 - l^2)).
From G. C. Greubel, Mar 31 2021: (Start)
a(n) = 8 * binomial(n+1, 2)^2 * C(n)^2, where C(n) = A000108(n) (Catalan numbers).
G.f.: 8*x*Hypergeometric2F1([3/2, 3/2], [1], 16*x) = (16/pi)*(x/(1-16*x)^2)*( 2*E(16*x) - (1-16*x)*K(16*x) ), where E(x) and K(x) are elliptic functions. (End)
D-finite with recurrence (n-1)^2*a(n) +(n^2-52*n+64)*a(n-1) -68*(2*n -3)^2*a(n-2)=0. - R. J. Mathar, Feb 27 2023
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MAPLE
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A254408:= n-> 2*( n*binomial(2*n, n) )^2; seq(A254408(n), n=0..30); # G. C. Greubel, Mar 31 2021
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MATHEMATICA
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a[n_] := 2*n^2*Binomial[2*n, n]^2; Table[a[n], {n, 0, 20}]
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PROG
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(PARI) a(n) = 2*n^2*binomial(2*n, n)^2 \\ Charles R Greathouse IV, May 10 2016
(Magma) [(4*Binomial(n+1, 2)*Catalan(n))^2/2: n in [0..30]]; // G. C. Greubel, Mar 31 2021
(Sage) [(4*binomial(n+1, 2)*catalan_number(n))^2/2 for n in (0..30)] # G. C. Greubel, Mar 31 2021
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CROSSREFS
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Cf. A000108, A000984, A002894.
Sequence in context: A187289 A187191 A054607 * A132592 A034977 A065141
Adjacent sequences: A254405 A254406 A254407 * A254409 A254410 A254411
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KEYWORD
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easy,nonn
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AUTHOR
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Jean-François Alcover, Jan 30 2015
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STATUS
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approved
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