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A082758
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Sum of the squares of the trinomial coefficients (A027907).
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6
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1, 3, 19, 141, 1107, 8953, 73789, 616227, 5196627, 44152809, 377379369, 3241135527, 27948336381, 241813226151, 2098240353907, 18252025766941, 159114492071763, 1389754816243449, 12159131877715993, 106542797484006471
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OFFSET
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0,2
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COMMENTS
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a(n) = T(2*n,2*n), the coefficient of x^(2*n) in (1+x+x^2)^(2*n), where T is the trinomial triangle A027907; Integral representation : a(n) = 1/Pi * Integral[(1+2*x)^(2*n)/Sqrt[1-x^2],{x,-1, 1}], i.e. a(n) is the moment of order 2n of the random variable 1+2X, where the distribution of X is an arcsin law on the interval (-1,1). - N-E. Fahssi, Jan 22 2008
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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a(n) = sum(k=0..2n+1, T(n, k)^2), where T(n, k) are trinomial coefficients (A027907).
a(n) = sum(k=0..n, binomial(2*n-k, k)*binomial(2*n, k)). - Benoit Cloitre, Jul 30 2003
G.f.: (1/sqrt(1+2*x-3*x^2)+1/sqrt(1-2*x-3*x^2))/2 (with interpolated zeros) - Paul Barry, Jan 04 2005
a(n) = sum(k=0..n, C(2*n,2*k)*C(2*k,k)) = sum(k=0..n, C(n+k,2k)*C(2*n,n+k)). [Paul Barry, Dec 16 2008]
a(n) = sum(k=0..n, binomial(n,k)^2*binomial(2*n,n)/binomial(2*k,k)). - Paul D. Hanna, Sep 29 2012
Recurrence: n*(2*n-1)*a(n) = (14*n^2+n-12)*a(n-1) + 3*(14*n^2-71*n+78)*a(n-2) - 27*(n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 3^(2*n+1/2)/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 14 2012
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MATHEMATICA
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Table[Sum[(-1)^(i)*Binomial[2*n, i]*Binomial[4*n-3*i-1, 2*n-3*i], {i, 0, 2*n/3}], {n, 0, 25}] (* From Adi Dani, Jul 03 2011 *)
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PROG
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(PARI) a(n)={local(v=Vec((1+x+x^2)^n)); sum(k=1, #v, v[k]^2); }
(PARI) a(n)=sum(k=0, n, binomial(2*n-k, k)*binomial(2*n, k));
(Maxima) makelist(sum(binomial(2*n-k, k)*binomial(2*n, k), k, 0, n), n, 0, 40);
(PARI) {a(n)=sum(k=0, n, binomial(n, k)^2*binomial(2*n, n)/binomial(2*k, k))} \\ Paul D. Hanna, Sep 29 2012
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CROSSREFS
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Bisection of A002426.
Sequence in context: A027314 A025571 A221266 * A110525 A058859 A095002
Adjacent sequences: A082755 A082756 A082757 * A082759 A082760 A082761
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KEYWORD
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nonn,easy,changed
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AUTHOR
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Emanuele Munarini, May 21 2003
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STATUS
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approved
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