|
%I #33 Jul 13 2016 11:39:21
%S 1,4,20,112,660,4004,24752,155040,980628,6249100,40060020,258048960,
%T 1668903600,10829900592,70480305440,459823234112,3006465218196,
%U 19694758782300,129235131438140,849311959095600,5589126007740660,36825913869817380,242910890642347200
%N a(n) = [x^n] (1 + x)/(1 - x)^(2*n+1).
%F a(n) = binomial(3*n, n) + binomial(3*n-1, n-1).
%F G.f.: (2*G(x) - 1) / (3 - 2*G(x)), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
%F 2*n*(2*n - 1)*a(n) - 3*(3*n - 1)*(3*n - 2)*a(n-1)=0. - _R. J. Mathar_, Jul 28 2013
%F a(n) = Sum_{r = 0..n-1} C(n-1,r)*C(2*n,r) + Sum_{r = 0..n} C(n,r)*C(2*n,n + r) - _J. M. Bergot_, Mar 18 2014
%F From _Peter Bala_, Jul 12 2016: (Start)
%F For n >= 1, a(n) = 4*binomial(3*n - 1, n - 1) = 4*A025174(n).
%F a(n) = [x^n]( 1/C(-x)^4 )^n, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. Cf. A224274. (End)
%e G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 660*x^4 + 4004*x^5 + ...
%e where a(n) equals the coefficient of x^n in (1+x)/(1-x)^(2*n+1)
%e and forms the main diagonal in the following table of coefficients:
%e (1+x)/(1-x)^1: [1, 2, 2, 2, 2, 2, 2, 2, 2, ...];
%e (1+x)/(1-x)^3: [1, 4, 9, 16, 25, 36, 49, 64, 81, ...];
%e (1+x)/(1-x)^5: [1, 6, 20, 50, 105, 196, 336, 540, ...];
%e (1+x)/(1-x)^7: [1, 8, 35, 112, 294, 672, 1386, 2640, ...];
%e (1+x)/(1-x)^9: [1, 10, 54, 210, 660, 1782, 4290, 9438, ...];
%e (1+x)/(1-x)^11:[1, 12, 77, 352, 1287, 4004, 11011, 27456, ...];
%e (1+x)/(1-x)^13:[1, 14, 104, 546, 2275, 8008, 24752, 68952, ...];
%e (1+x)/(1-x)^15:[1, 16, 135, 800, 3740, 14688, 50388, 155040, ...]; ...
%e Related series is G(x) = 1 + x*G(x)^3, which begins:
%e G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 +...+ A001764(n)*x^n +...
%t Join[{1},Table[Binomial[3n,n]+Binomial[3n-1,n-1],{n,30}]] (* _Harvey P. Dale_, Jan 15 2015 *)
%o (PARI) {a(n)=binomial(3*n,n)+binomial(3*n-1,n-1)}
%o (PARI) {a(n)=2*binomial(3*n-1, n) - 0^n}
%o (PARI) {a(n)=polcoeff((1+x)/(1-x+x*O(x^n))^(2*n+1),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A052227, A001764, A000108, A025174, A224274.
%K nonn,easy
%O 0,2
%A _Paul D. Hanna_, Jul 22 2013
|
