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A027277
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a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(2*n-k,k).
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1
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1, 3, 13, 67, 375, 2189, 13089, 79479, 487833, 3018355, 18792303, 117589689, 738844719, 4658460165, 29458662005, 186761788579, 1186655988771, 7554520173441, 48176764031385, 307706150625855, 1968040844127793, 12602972755261195, 80798365998084795, 518536437750443773
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OFFSET
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0,2
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COMMENTS
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Previous name was: a(n) = self-convolution of row n of array T given by A026568.
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LINKS
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FORMULA
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a(n) = hypergeom([1/2, -n, 1/2-n], [1, -2*n], -16) for n>=1.
a(n) = (2*n*(4*n-5)*(-9+4*n)*(-7+4*n)*a(n-3) - (4*n-5)*(50*n^3-175*n^2+152*n-9)* a(n-2) + (80*n^3-260*n^2+198*n-27)*(n-1)*a(n-1)) / (n*(n-1)*(-9+4*n)*(-1+2*n)) for n>=3. (End)
a(n) ~ sqrt(5 + 13/sqrt(17)) * ((9 + sqrt(17))/2)^n / (4*sqrt(Pi*n)). - Vaclav Kotesovec, May 14 2016
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MAPLE
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a := n -> add(binomial(2*k, k)*binomial(2*n-k, k), k=0..n):
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MATHEMATICA
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Table[Sum[Binomial[2k, k] Binomial[2n-k, k], {k, 0, n}], {n, 0, 30}] (* Michael De Vlieger, May 14 2016 *)
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PROG
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(PARI) vector(30, n, n--; b=binomial; sum(k=0, n, b(2*k, k)*b(2*n-k, k)) ) \\ G. C. Greubel, May 23 2017, modified Aug 03 2019
(Magma) B:=Binomial; [(&+[B(2*k, k)*B(2*n-k, k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Aug 03 2019
(Sage) b=binomial; [sum(b(2*k, k)*b(2*n-k, k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Aug 03 2019
(GAP) B:=Binomial;; List([0..30], n-> Sum([0..n], k-> B(2*k, k)*B(2*n-k, k) )); # G. C. Greubel, Aug 03 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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