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 A082758 Sum of the squares of the trinomial coefficients (A027907). 15
 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 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 77. (In the integral formula a left bracket is missing for the cosine argument.) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Jelena Đokić, Olga Bodroža-Pantić, and Ksenija Doroslovački, A spanning union of cycles in rectangular grid graphs, thick grid cylinders and Moebius strips, Transactions on Combinatorics (2023) Art. 27132. Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016. Veronika Irvine, Stephen Melczer, and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018. 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 From Peter Bala, Mar 16 2018: (Start) a(n) = sqrt(-3)^(2*n)*P(2*n,-1/sqrt(-3)), where P(n,x) is the Legendre polynomia1 of degree n. a(n) = 1/C(2*n,n)*Sum_{k = 0..n} C(n,k)*C(n+k,k)*C(2*n+2*k,n+k)*(-3)^(n-k). Cf. A273055. (End) From Wolfdieter Lang, Apr 19 2018 : (Start) a(n) = (2/Pi)*Integral_{phi=0..Pi/2} ( sin(3*phi))/sin(phi))^(2*n) [Comtet, p. 77, q=3, n=k -> 2*n] = (2/Pi)*Integral_{x=0..2} (x^2 - 1)^(2*n)/sqrt(4-x^2) (with x = 2*cos(phi)). See also the integral of the above comment. a(n) = 3^(2*n)*Sum_{k=0..2*n} binomial(2*n, k)*binomial(2*k, k)*(-1/3)^k = 3^(2*n)*hypergeometric([-2*n, 1/2], [1], 4/3) = (-3)^n*LegendreP(2*n, 1/sqrt(-3)))). (End) From Peter Bala, Apr 03 2022: (Start) Conjecture: a(n) = [x^n] ( (1 + x + x^3 + x^4)/(1 - x)^2 )^n. If the conjecture is true then the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Calculation suggests that the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for primes p >= 5 and positive integers n and k. Column 1 of A337389. (End) 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([next(A082758) for _ in range(20)]) # Peter Luschny, May 16 2016 CROSSREFS Cf. A002426, A027907, A273055, A337389. Sequence in context: A027314 A025571 A221266 * A110525 A331716 A058859 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 February 9 07:25 EST 2023. Contains 360153 sequences. (Running on oeis4.)