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A082758 Sum of the squares of the trinomial coefficients (A027907). 10
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

REFERENCES

Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

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 A291964

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 November 21 17:20 EST 2017. Contains 295004 sequences.