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A348899 a(n) = 332640*4^n*Gamma(n + 1/2)/(sqrt(Pi)*Gamma(n + 7)); super ballot numbers, row 5 of A135573.

%I A348899 #36 Dec 24 2024 22:11:36
%S A348899 462,132,99,110,154,252,462,924,1980,4488,10659,26334,67298,177100,
%T A348899 478170,1320660,3721860,10680120,31150350,92205036,276615108,
%U A348899 840090328,2580277436,8007757560,25090973688,79319852304,252832029219,812127124158,2627470107570,8558045493228
%N A348899 a(n) = 332640*4^n*Gamma(n + 1/2)/(sqrt(Pi)*Gamma(n + 7)); super ballot numbers, row 5 of A135573.
%F A348899 Let A[c, k](n) = c*4^n*Gamma(n + 1/2)/(sqrt(Pi)*Gamma(n + k)). Then
%F A348899   A[1, 1](n) = A000984(n).
%F A348899   A[3!, 3](n) = A007054(n).
%F A348899   A[5!*7, 5](n) = A348893(n).
%F A348899   A[7!*66, 7](n) = a(n).
%F A348899   A[c, k](n) ~ -c*2^(2*n - 1)*(k^2 - k - 2*n + 1/4)/(n^(k + 1/2)*sqrt(Pi)).
%F A348899 O.g.f.: ((2048*x^5 - 1686*x^4 + 765*x^3 - 178*x^2 + 21*x - 1)*sqrt(1 - 4*x) - 3496*x^5 + 2934*x^4 - 1083*x^3 + 218*x^2 - 23*x + 1)/(sqrt(1 - 4*x)*(1 + sqrt(1 - 4*x))*x^5).
%F A348899 E.g.f.: 1024*exp(2*x)*((-x^5 - 3/4*x^4 - 41/64*x^3 - 123/256*x^2 - 9/32*x - 15/128)*BesselI(1, 2*x) + BesselI(0, 2*x)*x*(x^4 + 1/2*x^3 + 27/64*x^2 + 9/32*x + 15/128))/x^5.
%F A348899 a(n) = Integral_{x=0..4} x^n*(4-x)^(11/2)/(2*Pi*sqrt(x)). This is the integral representation as the n-th moment of a positive function on [0, 4]. The representation is unique.
%F A348899 a(n) = 4^(n + 6)*hypergeom([13/2, 1/2 - n], [15/2], 1) / (13*Pi).
%F A348899 D-finite with recurrence (n+6)*a(n) +2*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Jul 27 2022
%F A348899 From _Peter Bala_, Mar 11 2023: (Start)
%F A348899 a(n) = 332640*(2*n)!/(n!*(n + 6)!).
%F A348899 a(n) = Sum_{k = 0..5} (-1)^k*4^(5-k)*binomial(n,k)*Catalan(n+k), where Catalan(n) = A000108(n). Thus a(n) is an integer for all n.
%F A348899 a(n) is odd if n = 2^k - 6, k >= 3, otherwise a(n) is even. (End)
%F A348899 From _Amiram Eldar_, Mar 28 2023: (Start)
%F A348899 Sum_{n>=0} 1/a(n) = 101/3465 + 52*Pi/(6561*sqrt(3)).
%F A348899 Sum_{n>=0} (-1)^(n+1)/a(n) = 8573/54140625 + 104*log(phi)/(78125*sqrt(5)), where phi is the golden ratio (A001622). (End)
%p A348899 a := n -> 332640*4^n*GAMMA(n + 1/2)/(sqrt(Pi)*GAMMA(n + 7));
%p A348899 seq(a(n), n = 0..29);
%t A348899 a[n_] := 4^(n + 6) Hypergeometric2F1[13/2, 1/2 - n, 15/2, 1] / (13 Pi);
%t A348899 Table[a[n], {n, 0, 29}]
%t A348899 Array[332640*4^#*Gamma[# + 1/2]/(Sqrt[Pi]*Gamma[# + 7]) &, 30, 0] (* _Michael De Vlieger_, Nov 02 2021 *)
%Y A348899 Row 5 of array A135573.
%Y A348899 Cf. A000108, A000984, A001622, A007054, A348893.
%K A348899 nonn
%O A348899 0,1
%A A348899 _Peter Luschny_, Nov 02 2021