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A082758 Sum of the squares of the trinomial coefficients (A027907). 9
1, 3, 19, 141, 1107, 8953, 73789, 616227, 5196627, 44152809, 377379369, 3241135527, 27948336381, 241813226151, 2098240353907, 18252025766941, 159114492071763, 1389754816243449, 12159131877715993, 106542797484006471 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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_{x=-1..1} ((1+2*x)^(2*n)/sqrt(1-x^2)), 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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

StackExchange, The asymptotic behavior of an integral, 2015

FORMULA

a(n) = Sum_{k=0..2n} 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} binomial(2*n,2*k)*binomial(2*k,k) = Sum_{k=0..n} binomial(n+k,2k)*binomial(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

a(n) = GegenbauerC(2*n, -2*n, -1/2). - Peter Luschny, May 07 2016

From Peter Luschny, May 15 2016: (Start)

a(n) = ((9-9*n)*(4*n-1)*(2*n-3)*a(n-2)+(4*n-3)*(20*n^2-30*n+7)*a(n-1))/(n*(2*n-1)*(4*n-5)) for n>=2.

a(n) = hypergeom([1/2-n, -n], [1], 4). (End)

a(n) = A002426(2*n). - Michael Somos, Jan 08 2017

EXAMPLE

G.f. = 1 + 3*x + 19*x^2 + 141*x^3 + 1107*x^4 + 8953*x^5 + 73789*x^6 + ...

MAPLE

a := n -> simplify(GegenbauerC(2*n, -2*n, -1/2)):

seq(a(n), n=0..19); # Peter Luschny, May 07 2016

MATHEMATICA

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}] (* Adi Dani, Jul 03 2011 *)

Table[Hypergeometric2F1[1/2-n, -n, 1, 4], {n, 0, 19}] (* Peter Luschny, May 15 2016 *)

a[ n_] := SeriesCoefficient[ (1 - 2 x - 3 x^2)^(-1/2), {x, 0, 2 n}]; (* Michael Somos, Jan 08 2017 *)

PROG

(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

(Sage)

def A():

    a, b, n = 1, 3, 1

    yield a

    while True:

        yield b

        n += 1

        a, b = b, ((9-9*n)*(4*n-1)*(2*n-3)*a+(4*n-3)*(20*n^2-30*n+7)*b)//(n*(2*n-1)*(4*n-5))

A082758 = A()

print([A082758.next() for _ in range(20)]) # Peter Luschny, May 16 2016

CROSSREFS

Cf. A002426.

Sequence in context: A027314 A025571 A221266 * A110525 A058859 A095002

Adjacent sequences:  A082755 A082756 A082757 * A082759 A082760 A082761

KEYWORD

nonn,easy

AUTHOR

Emanuele Munarini, May 21 2003

STATUS

approved

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Last modified May 27 15:22 EDT 2017. Contains 287207 sequences.