cp's OEIS Frontend

A386517 a(n) = 60*binomial(3*n,n)/(n+2).

%I A386517 #43 Aug 28 2025 13:39:46
%S A386517 30,60,225,1008,4950,25740,139230,775200,4412826,25564500,150225075,
%T A386517 893246400,5364333000,32489701776,198225859050,1217179149120,
%U A386517 7516163045490,46645481326500,290779045735815,1819954198062000,11432303197651350,72050701049642700,455457919954401000
%N A386517 a(n) = 60*binomial(3*n,n)/(n+2).
%H A386517 Paolo Xausa, <a href="/A386517/b386517.txt">Table of n, a(n) for n = 0..1000</a>
%F A386517 a(n) = 60*A005809(n)/(n+2).
%F A386517 O.g.f.: (-i*(x + 1)*x*sqrt(3)*t - 4*sqrt(x) + 15*x^(3/2) + 81*x^(5/2))*(4*t - 12*i*sqrt(3)*sqrt(x))^(1/3) + (i*(x + 1)*x*sqrt(3)*t - 4*sqrt(x) + 15*x^(3/2) + 81*x^(5/2))*(4*t + 12*i*sqrt(3)*sqrt(x))^(1/3) + 8*t*sqrt(x))/(4*t*x^(5/2)), for t = sqrt(4-27*x), and i = sqrt(-1) the imaginary unit.
%F A386517 O.g.f.: 30*hypergeom([1/3, 2/3, 2], [1/2, 3], 27*x/4), that, denoted by h(x),satisfies
%F A386517   -270 - 540*x + 675*x^2 + 1728*x^3 + 9*(1 + 10*x^2 + 20*x^3)*h(x) - 6*x^2*h(x)^2 + x^4*h(x)^3 = 0.
%F A386517 E.g.f.: 30*hypergeom([1/3, 2/3, 2], [1/2, 1, 3], 27*x/4).
%F A386517 a(n) = Integral_{x=0..27/4} x^n*W(x)*dx, where, for S = sqrt(27 - 4*x),
%F A386517 W(x) = ((-6^(1/3)*(9 + sqrt(3)*S)^(2/3)*(sqrt(3) - 3*S) + 6*3^(1/6)*(1 + sqrt(3)*S)*x^(1/3) + 6^(1/3)*(9 + sqrt(3)*S)^(2/3)*(-sqrt(3) + S)*x + 2*3^(1/6)*(3 + sqrt(3)*S)*x^(4/3))*2^(1/3))/(8*Pi*(9 + sqrt(3)*S)^(1/3)*x^(2/3)).
%F A386517 W(x) is a positive function on x = (0,27/4), is singular at x=0, and tends to zero at x = 27/4. Thus a(n) is a positive definite sequence. This representation is unique as W(x) is the solution of the Hausdorff moment problem.
%t A386517 A386517[n_] := 60*Binomial[3*n, n]/(n + 2); Array[A386517, 25, 0] (* _Paolo Xausa_, Aug 28 2025 *)
%o A386517 (PARI) a(n) = 60*binomial(3*n,n)/(n+2);
%o A386517 vector(23,n,a(n-1)) \\ _Joerg Arndt_, Aug 26 2025
%Y A386517 Cf. A005809.
%K A386517 nonn,new
%O A386517 0,1
%A A386517 _Karol A. Penson_, Aug 25 2025