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%I #47 Jan 10 2022 09:20:35
%S 1,2,8,35,160,752,3599,17446,85376,420884,2087008,10398016,52010479,
%T 261021854,1313707256,6628095035,33512880640,169768235840,
%U 861450392708,4377796514152,22277498220160,113502759811000,578931209245760,2955873376166144,15105883318474991
%N a(n) = Sum_{k = 0..n} binomial(n,k)*trinomial(n,k), where trinomial(n,k) = trinomial coefficients.
%C Central coefficients of A115990. - _Paul Barry_, Feb 25 2011
%H Vincenzo Librandi, <a href="/A082759/b082759.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k = 0..n} C(n+k, n-k)*C(n, k). - _Benoit Cloitre_, Jun 20 2003
%F 2*n*(2*n - 1)*(38*n - 53)*a(n) + ( - 760*n^3 + 1820*n^2 - 1252*n + 252)*a(n - 1) - 8*(n - 1)*(19*n^2 - 36*n + 9)*a(n - 2) - 3*(38*n - 15)*(n - 1)*(n - 2)*a(n - 3) = 0. - _Vladeta Jovovic_, Jul 15 2004
%F a(n) = Sum_{k = 0..n} C(2*n - k, k)*C(n, k). - _Paul Barry_, Jan 20 2005
%F a(n) ~ c * d^n / sqrt(Pi*n), where d = 5.21913624874158651... = (((1261 + 57*sqrt(57))^(2/3) + 112 + 10*(1261 + 57*sqrt(57))^(1/3))/(6*(1261 + 57*sqrt(57))^(1/3))) is the real root of the equation 4*d^3 - 20*d^2 - 4*d - 3 = 0 and c = 0.79036380822702870439029... = 1/114*sqrt(57)*sqrt((9747 + 57*sqrt(57))^(1/3)*(2*(9747 + 57*sqrt(57))^(2/3) + 912 + 57*(9747 + 57*sqrt(57))^(1/3)))/((9747 + 57*sqrt(57))^(1/3)) is the positive real root of the equation 1216*c^6 - 912*c^4 + 100*c^2 - 3 = 0. - _Vaclav Kotesovec_, Oct 24 2012 (updated Oct 16 2016, following a suggestion of _Michael Somos_)
%F G.f.: A(x) = x*B'(x)/B(x), where B(x) satisfies B(x) = x*(1 + 2*B(x) + 2*B(x)^2 + B(x)^3). - _Vladimir Kruchinin_, Jan 14 2015
%F a(n) = Sum_{k = 0..n} (-1)^k*C(n, k)*C(3*n - 2*k, n - k). - _Peter Bala_, Jul 13 2016
%F G.f. y = A(x) satisfies 0 = 1 + y*(3-2*x) + y^3*(-4+20*x+4*x^2+3*x^3). - _Michael Somos_, Oct 15 2016
%F From _Peter Bala_, Jan 09 2022: (Start)
%F a(n) = [x^n] (1 + 2*x + 2*x^2 + x^3)^n.
%F The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)
%e G.f. = 1 + 2*x + 8*x^2 + 35*x^3 + 160*x^4 + 752*x^5 + 3599*x^6 + 17446*x^7 + ...
%t Table[Sum[Binomial[2 n - k, k] Binomial[n, k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 24 2012; typo fixed by _Vincenzo Librandi_, May 07 2013 *)
%o (PARI) a(n)=sum(k=0,n,binomial(n+k,n-k)*binomial(n,k))
%Y Cf. A037011, A106228, A115990.
%K nonn,easy
%O 0,2
%A _Emanuele Munarini_, May 21 2003
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