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Search: id:A132310
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| A132310 |
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a(n) = 3^n*Sum_{ k=0..n } binomial(2*k,k)/3^k. |
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+0
12
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| 1, 5, 21, 83, 319, 1209, 4551, 17085, 64125, 240995, 907741, 3428655, 12990121, 49370963, 188229489, 719805987, 2760498351, 10615101273, 40920439119, 158106581157, 612166272291, 2374756691313, 9228369037659, 35918537840577
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Simpler definition from N. J. A. Sloane (njas(AT)research.att.com), Jan 21 2009. Colin Mallows and I studied this sequence on Feb 21 1981 in connection with integration over a regular (solid) hexagon.
Hankel transform is A137717. [From Paul Barry (pbarry(AT)wit.ie), Apr 26 2009]
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FORMULA
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a(n) = C(2n,n) * Sum_{k=0..2n} trinomial(n,k)/C(2n,k) where trinomial(n,k) = [x^k] (1 + x + x^2)^n, where [x^k] denotes "coefficient of x^k in ...".
G.f.: A(x) = 1/sqrt(1 - 10*x + 33*x^2 - 36*x^3).
a(n) = Sum_{k=0..2n} trinomial(n,k) * k!*(2*n-k)! / (n!)^2 .
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EXAMPLE
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a(1) = C(2,1)*(1/1 + 1/2 + 1/1) = 2*(5/2) = 5;
a(2) = C(4,2)*(1/1 + 2/4 + 3/6 + 2/4 + 1/1) = 6*(7/2) = 21;
a(3) = C(6,3)*(1/1 + 3/6 + 6/15 + 7/20 + 6/15 + 3/6 + 1/1) = 20*(83/20) = 83.
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PROGRAM
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(PARI) {a(n)=binomial(2*n, n)*sum(k=0, 2*n, polcoeff((1+x+x^2)^n, k)/binomial(2*n, k) )} (PARI) {a(n)=sum(k=0, 2*n, polcoeff((1+x+x^2)^n, k) * k!*(2*n-k)! / (n!)^2 )}
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CROSSREFS
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Sequence in context: A094834 A147504 A026017 this_sequence A083319 A146041 A146585
Adjacent sequences: A132307 A132308 A132309 this_sequence A132311 A132312 A132313
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 18 2007
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