A384668 a(n) = 12 * (5*n+2)! / ((3*n+1)! * (2*n+2)!).

12, 105, 1584, 29172, 596904, 13037895, 297748800, 7023149820, 169774618104, 4183919862474, 104722807600320, 2654939113240050, 68033328627480804, 1759318006963275528, 45853277234783179392, 1203249937243079847660, 31764232607604306053400, 842982010030680328418706

Offset: 0

Author

Karol A. Penson, Jun 06 2025

Keywords

• nonn

Links

• Paolo Xausa, Table of n, a(n) for n = 0..650

Crossrefs

Cf. A384585, A002293, A000260, A001450.

Programs

Mathematica

A384668[n_] := 12*(5*n + 2)!/((3*n + 1)!*(2*n + 2)!);
Array[A384668, 20, 0] (* Paolo Xausa, Jun 12 2025 *)

Formula

O.g.f.: 12*hypergeom([3/5, 4/5, 1, 6/5, 7/5], [2/3, 4/3, 3/2, 2], (3125*x)/108).

E.g.f.: 12*hypergeom([3/5, 4/5, 6/5, 7/5], [2/3, 4/3, 3/2, 2], (3125*x)/108).

O.g.f. denoted by h(x), satisfies the algebraic equation of order 10:

1889568 - 6141096*x + 10628820*x^2 - 59049*x^3 + (-2834352*x^3 + 4861701*x^2 - 2834352*x - 157464)*h(x) + 13122*x*(14*x^3 - 77*x^2 + 124*x + 30)*h(x)^2 - 4374*x^2*(14*x^2 + 94*x + 99)*h(x)^3 + 729*x^3*(50*x^2 + 32*x + 377)*h(x)^4 - 243*x^4*(11*x^2 - 40*x + 456)*h(x)^5 - 243*x^5*(8*x - 121)*h(x)^6 + 54*x^6*(2*x - 95)*h(x)^7 + 567*x^7*h(x)^8 - 36*x^8*h(x)^9 + x^9*h(x)^10 = 0.

a(n) = Integral_{x=0..3125/108} x^n*W(x)*dx, where W(x) = W1(x)+W2(x)+W3(x)+W4(x), with

W1(x) = (3*sqrt(5)*csc(Pi/5)*sin(Pi/10)*hypergeom([-2/5, 1/10, 4/15, 14/15], [1/5, 2/5, 4/5], (108*x)/3125))/(2*Pi*x^(2/5)),

W2(x) = (6*sqrt(5)*csc((2*Pi)/5)*sin((3*Pi)/10)*hypergeom([-1/5, 3/10, 7/15, 17/15], [2/5, 3/5, 6/5], (108*x)/3125))/(5*Pi*x^(1/5)),

W3(x) = -(24*sqrt(5)*csc((2*Pi)/5)*sin((3*Pi)/10)*x^(1/5)*hypergeom([1/5, 7/10, 13/15, 23/15], [4/5, 7/5, 8/5], (108*x)/3125))/(125*Pi), and

W4(x) = -(33*sqrt(5)*csc(Pi/5)*sin(Pi/10)*x^(2/5)*hypergeom([2/5, 9/10, 16/15, 26/15], [6/5, 8/5, 9/5], (108*x)/3125))/(1250*Pi).

This integral representation is unique as it is the solution of the Hausdorff power moment of the function W(x). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 3125/108. Therefore a(n) is a positive definite sequence.