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A066380
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a(n) = Sum_{k=0..n} binomial(3*n,k).
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7
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1, 4, 22, 130, 794, 4944, 31180, 198440, 1271626, 8192524, 53009102, 344212906, 2241812648, 14637774688, 95786202688, 628002401520, 4124304597834, 27126202533252, 178651732923346, 1178005033926998, 7776048412324714
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OFFSET
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0,2
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REFERENCES
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R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 425.
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LINKS
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FORMULA
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a(n) ~ C(3n, n)(2 - 4/n + O(1/n^2)).
G.f.: (1-g)/((3*g-1)*(2*g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
a(0)=1, a(n) = 8*a(n-1) - (5*n^2+n-2)*(3*n-3)!/((2*n-1)!*n!). - Tani Akinari, Sep 02 2014
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+1,n-2*k). - Seiichi Manyama, Apr 09 2024
a(n) = binomial(1+3*n, n)*hypergeom([1, (1-n)/2, -n/2], [1+n, 3/2+n], 1). - Stefano Spezia, Apr 09 2024
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MAPLE
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MATHEMATICA
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Table[Sum[Binomial[3 n, k], {k, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, May 27 2013 *)
a[n_] := 8^n - (2*n)/(n+1)*Binomial[3*n, n]*Hypergeometric2F1[1, -2*n+1, n+2, -1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 02 2013 *)
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PROG
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(PARI) { for (n=0, 150, a=0; for (k=0, n, a+=binomial(3*n, k)); write("b066380.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 12 2010
(Maxima) a[0]:1$ a[n]:=8*a[n-1]-(5*n^2+n-2)*(3*n-3)!/((2*n-1)!*n!)$ makelist(a[n], n, 0, 200); /* Tani Akinari, Sep 02 2014 */
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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