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%I #12 Jul 25 2023 07:27:25
%S 1,1,7,61,603,6443,72517,846995,10170685,124780525,1557347467,
%T 19710577873,252386341335,3263626001751,42558647522697,
%U 559032393114023,7390085367865081,98242108076244665,1312529311579827631,17613845480108029957,237322279651518516019
%N G.f. satisfies A(x) = 1 - x*A(x)*(1 - 2*A(x)^3).
%F a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(n+3*k+1,n) / (n+3*k+1).
%F D-finite with recurrence 3*n*(3*n-1)*(3*n+1)*a(n) +(-566*n^3 +1335*n^2 -1105*n +312)*a(n-1) +3*(943*n^3 -5739*n^2 +11016*n -6748)*a(n-2) +18*(-250*n^3 +2499*n^2 -8233*n +8938)*a(n-3) +27*(n-4)*(31*n^2 -314*n +735)*a(n-4) +81*(10*n -51)*(n-4) *(n-5)*a(n-5) +243*(n-5) *(n-6)*(n-4)*a(n-6)=0. - _R. J. Mathar_, Jul 25 2023
%p A364430 := proc(n)
%p (-1)^n*add((-2)^k* binomial(n,k) * binomial(n+3*k+1,n) / (n+3*k+1),k=0..n) ;
%p end proc:
%p seq(A364430(n),n=0..70); # _R. J. Mathar_, Jul 25 2023
%o (PARI) a(n) = (-1)^n*sum(k=0, n, (-2)^k*binomial(n, k)*binomial(n+3*k+1, n)/(n+3*k+1));
%Y Cf. A001003, A153232.
%Y Cf. A364431, A364432, A364437.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Jul 24 2023
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