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

0,2

Links

Table of n, a(n) for n=0..17.

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.

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 *)

Prog

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

Crossrefs

Cf. A004355.

Sequence in context: A277464 A258300 A276752 * A052172 A058783 A203769

Adjacent sequences: A380604 A380605 A380606 * A380608 A380611 A380612

Keyword

nonn,new

Author

Karol A. Penson, Jan 28 2025

Status

approved