A380607 a(0) = 1, a(n) = 5*binomial(6*(n-1),n-1), for n > 0.

1, 5, 30, 330, 4080, 53130, 712530, 9738960, 134891640, 1886744970, 26589681300, 376970137830, 5370413979840, 76816421507280, 1102478371452150, 15868672192650600, 228978369822304080, 3311260421942706570

Offset: 0

Author

Karol A. Penson, Jan 28 2025

Keywords

• nonn

Crossrefs

Cf. A004355.

Programs

Mathematica

CoefficientList[Series[5*z*HypergeometricPFQ[{1/6, 1/3, 1/2, 2/3, 5/6}, {1/5, 2/5, 3/5, 4/5}, (6^6*z)/5^5] + 1,{z,0,17}],z] (* Stefano Spezia, Jan 28 2025 *)

PARI

a(n) = 0^n+binomial(6*(n-1),n-1)*5 \\ Thomas Scheuerle, Jan 29 2025

Formula

a(n) = 5*A004355(n-1), for n>=1.

G.f.: h(z) = 5*z*hypergeom([1/6, 1/3, 1/2, 2/3, 5/6], [1/5, 2/5, 3/5, 4/5], (6^6*z)/5^5) + 1

satisfies: 15625*z^6 - 75000*z^5 + 140625*z^4 - 125000*z^3 + 46875*z^2 + 46656*z - 3125 + (75000*z^5 - 281250*z^4 + 375000*z^3 - 187500*z^2 - 279936*z + 18750)*h(z) + (140625*z^4 - 375000*z^3 + 281250*z^2 + 699840*z - 46875)*h(z)^2 + (125000*z^3 - 187500*z^2 - 933120*z + 62500)*h(z)^3 + (46875*z^2 + 699840*z - 46875)*h(z)^4 + (-279936*z + 18750)*h(z)^5 + (46656*z - 3125)*h(z)^6 = 0.

a(n) = Integral_{x=0..sup} x^n*W(x), where sup = 6^6/5^5, with W(x) = (5^10)*sqrt(15)/((6^12)*sqrt(Pi) )*MeijerG([[],[-1,-9/5,-8/5,-7/5,-6/5]],[[-11/6,-5/3,-3/2,-4/3,-7/6],[]],x/(6^6/5^5)), n>0. In W(x) MeijerG is the Meijer G-function in Maple notation, which can be represented as the sum of five generalized hypergeometric functions of type 5F4. This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem on x = (0, sup). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is U-shaped, is singular at x = 0, with singularity x^(-1/6), and is singular at x = sup. W(x) has a minimum at x around x=11.