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A188648
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Binomial sums a(n) = Sum_{k=0..n} (binomial(2n-k,k))^2.
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5
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1, 2, 11, 63, 376, 2317, 14545, 92512, 594169, 3844787, 25027296, 163701327, 1075049011, 7083830648, 46812088751, 310118453573, 2058919125662, 13695571200353, 91254952276859, 608960974528058, 4069232436916151
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OFFSET
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0,2
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COMMENTS
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Bisection of A051286 (Whitney number of level n of the lattice of the ideals of the fence of order 2n). - Paul D. Hanna, Apr 07 2011
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LINKS
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FORMULA
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G.f.: 1/2*(1/sqrt(1-2*sqrt(x)-x-2*x*sqrt(x)+x^2) + 1/sqrt(1+2*sqrt(x)-x+2*x*sqrt(x)+x^2)).
Recurrence: (n-2)*n*(2*n - 1)*(48*n^2 - 192*n + 169)*a(n) = (576*n^5 - 4032*n^4 + 10212*n^3 - 11414*n^2 + 5457*n - 849)*a(n-1) + 5*(2*n - 3)*(48*n^4 - 288*n^3 + 565*n^2 - 399*n + 64)*a(n-2) + (576*n^5 - 4608*n^4 + 13668*n^3 - 18286*n^2 + 10521*n - 1896)*a(n-3) - (n-3)*(n-1)*(2*n - 5)*(48*n^2 - 96*n + 25)*a(n-4). - Vaclav Kotesovec, Mar 02 2014
a(n) ~ phi^(4*n + 2) / (2^(3/2) * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 02 2014, simplified Jan 13 2019
Conjecture: a(n) = hypergeom([-n,-n,n+1,n+1], [1/2,1/2,1], 1/16). - Velin Yanev, Oct 31 2019
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MATHEMATICA
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Table[Sum[Binomial[2n-k, k]^2, {k, 0, n}], {n, 0, 20}]
Table[DifferenceRoot[Function[{y, m}, {4 (-m + n)^2 (-1 - 2 m + 2 n)^2 y[m] + (-5 m^2 - 18 m^3 - 17 m^4 + 12 m n + 56 m^2 n + 68 m^3 n - 8 n^2 - 56 m n^2 - 100 m^2 n^2 + 16 n^3 + 64 m n^3 - 16 n^4) y[1 + m] + (1 + m)^2 (-m + 2 n)^2 y[2 + m] == 0, y[0] == 0, y[1] == 1}]][n + 1], {n, 0, 20}] (* Benedict W. J. Irwin, Nov 03 2016 *)
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PROG
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(Maxima) makelist(sum(binomial(2*n-k, k)^2, k, 0, n), n, 0, 20);
(PARI) {a(n) = sum(k=0, n, binomial(2*n-k, k)^2)} \\ Seiichi Manyama, Jan 13 2019
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CROSSREFS
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Sum_{k=0..n} (binomial(2n-k,k))^b: A122367(n) = A001519(n+1) (b=1), this sequence (b=2).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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