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A336495
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a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).
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4
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1, 1, 4, 34, 484, 9946, 270314, 9189776, 376223992, 18046839982, 993655820512, 61803730636506, 4287521490060780, 328324625277864008, 27514775912958768464, 2505202120094546731584, 246288599061132553970160, 26004541628560046316399382, 2935176211106696031739535696
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OFFSET
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0,3
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COMMENTS
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Number of Sylvester classes of n-packed words of degree n.
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LINKS
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FORMULA
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a(n) = ( (-1)^n / (n^2+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(n^2+1,k) * binomial((n+1)*n-k,n-k).
a(n) = (-1)^n*binomial(1 + n^2, n)*hypergeom[-n, 1 + n^2, 2 + (n - 1)*n, 2] / (1 + n^2). - Peter Luschny, Jul 26 2020
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial((n+1)*n-k,n-1-k) for n > 0. - Seiichi Manyama, Aug 08 2023
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MATHEMATICA
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a[n_] := ((-1)^n Binomial[1 + n^2, n] Hypergeometric2F1[-n, 1 + n^2, 2 + (n - 1) n, 2]) / (1 + n^2); Array[a, 19, 0] (* Peter Luschny, Jul 26 2020 *)
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PROG
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(PARI) a(n) = (-1)^n*sum(k=0, n, (-2)^k*binomial(n, k)*binomial(n^2+k+1, n)/(n^2+k+1));
(PARI) a(n) = (-1)^n*sum(k=0, n, (-2)^(n-k)*binomial(n^2+1, k)*binomial((n+1)*n-k, n-k))/(n^2+1);
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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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