cp's OEIS Frontend

A364372 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^3).

%I A364372 #17 Sep 09 2024 09:35:24
%S A364372 1,0,-1,3,-6,6,15,-107,349,-672,39,5835,-27654,75765,-95799,-279129,
%T A364372 2297970,-8377854,17663640,-996624,-177445221,888491025,-2551959604,
%U A364372 3337931168,10407149226,-87719805853,328682535695,-708428979213,15252552804,7616368090377,-38693979668535
%N A364372 G.f. satisfies A(x) = (1 + x) * (1 - x*A(x)^3).
%F A364372 a(n) = Sum_{k=0..n} (-1)^k * binomial(3*k+1,k) * binomial(3*k+1,n-k) / (3*k+1).
%F A364372 D-finite with recurrence 2*n*(2*n+1)*a(n) +(51*n-26)*(n-1)*a(n-1) +(279*n^2 -931*n +766)*a(n-2) +2*(413*n^2 -2127*n +2728)*a(n-3) +6*(75*n-244) *(3*n-11)*a(n-4) +9*(45*n-179) *(3*n-14)*a(n-5) +63*(3*n-14) *(3*n-17)*a(n-6) +12*(3*n-16) *(3*n-20)*a(n-7)=0. - _R. J. Mathar_, Jul 25 2023
%F A364372 From _Peter Bala_, Aug 25 2024: (Start)
%F A364372 Fifth-order recurrence: 2*(n-2)*(n-1)*n*(2*n+1)*a(n) + (n-2)*(n-1)*(31*n^2-27*n+6)*a(n-1) + 3*(n-2)*(3*n-5)*(12*n^2-19*n+2)*a(n-2) + 3*(3*n-8)*(18*n^3-69*n^2+63*n-2)*a(n-3) + 3*n*(3*n-11)*(12*n^2-49*n+42)*a(n-4) + 3*n*(n-1)*(3*n-10)*(3*n-14)*a(n-5) = 0 with a(0) = 1, a(1) = 0, a(2) = -1, a(3) = 3 and a(4) = -6.
%F A364372 The g.f. A(x) satisfies (1/x) * series_reversion(x/A(x)) = 1 - x^2 + 3*x^3 - 4*x^4 - 9*x^5 + 73*x^6 - ..., the g.f. of A364376. (End)
%p A364372 A364372 := proc(n)
%p A364372     add( (-1)^k*binomial(3*k+1,k) * binomial(3*k+1,n-k)/(3*k+1),k=0..n) ;
%p A364372 end proc:
%p A364372 seq(A364372(n),n=0..80); # _R. J. Mathar_, Jul 25 2023
%o A364372 (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(3*k+1, k)*binomial(3*k+1, n-k)/(3*k+1));
%Y A364372 Cf. A364371, A364373, A364376.
%Y A364372 Cf. A364336.
%K A364372 sign,easy
%O A364372 0,4
%A A364372 _Seiichi Manyama_, Jul 20 2023