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

%I A380607 #23 Feb 02 2025 08:48:44
%S A380607 1,5,30,330,4080,53130,712530,9738960,134891640,1886744970,
%T A380607 26589681300,376970137830,5370413979840,76816421507280,
%U A380607 1102478371452150,15868672192650600,228978369822304080,3311260421942706570
%N A380607 a(0) = 1, a(n) = 5*binomial(6*(n-1),n-1), for n > 0.
%F A380607 a(n) = 5*A004355(n-1), for n>=1.
%F A380607 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 A380607 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 A380607 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.
%t A380607 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 A380607 (PARI) a(n) = 0^n+binomial(6*(n-1),n-1)*5 \\ _Thomas Scheuerle_, Jan 29 2025
%Y A380607 Cf. A004355.
%K A380607 nonn
%O A380607 0,2
%A A380607 _Karol A. Penson_, Jan 28 2025