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%I #22 Feb 10 2024 14:04:44
%S 1,2,9,61,493,4371,41065,401563,4044097,41658044,436862457,4648331765,
%T 50057856881,544557984498,5975422922413,66059269445451,
%U 735064865871889,8226310738656892,92531697191189777,1045551973586825023,11862334695799444993
%N G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(x)^3)).
%H Seiichi Manyama, <a href="/A349289/b349289.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = Sum_{k=0..n} binomial(n+2*k,3*k) * binomial(4*k,k) / (3*k+1).
%F a(n) = F([1/4, 1/2, 3/4, (1+n)/2, (2+n)/2, -n], [1/3, 2/3, 2/3, 1, 4/3], -2^10/3^6) where F is the generalized hypergeometric function. - _Stefano Spezia_, Nov 13 2021
%F a(n) ~ sqrt(1 + 2*r) / (4 * 2^(1/6) * sqrt(3*Pi*(1-r)) * n^(3/2) * r^(n + 1/3)), where r = 0.0816785448577670972635343365300887975661075663022821172271... is the root of the equation 4^4 * r = 3^3 * (1-r)^3. - _Vaclav Kotesovec_, Nov 14 2021
%F D-finite with recurrence 81*n*(3*n-1)*(3*n+1)*a(n) +3*(243*n^3-7101*n^2+9986*n-3560)*a(n-1) +(-115027*n^3+514908*n^2-699869*n+269580)*a(n-2) +(-85543*n^3+1604715*n^2-6291692*n+6995280)*a(n-3) +(580211*n^3-6643158*n^2+23063299*n-23830944)*a(n-4) +(-33473*n^3-2231073*n^2+26352470*n-70945392)*a(n-5) +(-872129*n^3+17812344*n^2-119542699*n+264170868)*a(n-6) +(667171*n^3-14196243*n^2+100393472*n-236010000)*a(n-7) -6*(3*n-23)*(9948*n^2-147805*n+548868)*a(n-8) +4044*(3*n-26)*(n-8)*(3*n-22)*a(n-9)=0. - _R. J. Mathar_, Feb 10 2024
%p A349289 := proc(n)
%p add( binomial(n+2*k,3*k)*binomial(4*k,k)/(3*k+1),k=0..n) ;
%p end proc:
%p seq(A349289(n),n=0..50) ; # _R. J. Mathar_, Feb 10 2024
%t nmax = 20; A[_] = 0; Do[A[x_] = 1/((1 - x) (1 - x A[x]^3)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t Table[Sum[Binomial[n + 2 k, 3 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 20}]
%Y Cf. A002293, A007317, A199475, A346646, A349290, A349291, A349292, A349293.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 13 2021
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