A364432 - OEIS

%I #9 Jul 25 2023 07:35:13

%S 1,3,18,162,1728,20169,249318,3207600,42500700,576012060,7947785448,

%T 111269613006,1576658688480,22568473199358,325855352769588,

%U 4740157737123696,69405108247439676,1022070746845708740,15127922880893671704,224931239520535651464

%N G.f. satisfies A(x) = 1 + x*A(x)*(2 + A(x)^3).

%F a(n) = Sum_{k=0..n} 2^(n-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) +2*(-74*n^3 -375*n^2+ 665*n -252)*a(n-1) +12*(-337*n^3 +1941*n^2 -2984*n +1092)*a(n-2) +144*(-70*n^3 +861*n^2 -3347*n +4152)*a(n-3) +432*(n-4)*(31*n^2 -314*n +735)*a(n-4) -2592*(10*n-51) *(n-4)*(n-5)*a(n-5) +15552*(n-5)*(n-6) *(n-4)*a(n-6)=0. - _R. J. Mathar_, Jul 25 2023

%p A364432 := proc(n)

%p     add(2^(n-k)* binomial(n,k) * binomial(n+3*k+1,n) / (n+3*k+1),k=0..n) ;

%p end proc:

%p seq(A364432(n),n=0..70); # _R. J. Mathar_, Jul 25 2023

%o (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n, k)*binomial(n+3*k+1, n)/(n+3*k+1));

%Y Cf. A047891, A348793.

%Y Cf. A364430, A364431.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jul 24 2023