A380607 - OEIS

%I #28 Sep 24 2025 06:47:03

%S 1,5,30,330,4080,53130,712530,9738960,134891640,1886744970,

%T 26589681300,376970137830,5370413979840,76816421507280,

%U 1102478371452150,15868672192650600,228978369822304080,3311260421942706570,47975973011904098700,696292401504874983900

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

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

%F 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

%F 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.

%F 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.

%F a(n) ~ 64^(n-1) * 3^(6*n-11/2) / (5^(5*n-11/2) * sqrt(Pi*n)). - _Amiram Eldar_, Sep 24 2025

%t 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 *)

%o (PARI) a(n) = 0^n+binomial(6*(n-1),n-1)*5 \\ _Thomas Scheuerle_, Jan 29 2025

%Y Cf. A004355.

%K nonn,easy

%O 0,2

%A _Karol A. Penson_, Jan 28 2025