A390337 - OEIS

%I #18 Jan 03 2026 12:54:32

%S 1,3,17,103,643,4082,26193,169324,1100561,7183179,47037770,308839996,

%T 2032250927,13397582173,88463267588,584916078291,3872067079555,

%U 25659566239274,170200888987121,1129893804436178,7506551997419770,49904474883265592,331976969392263124

%N a(n) = Sum_{k=0..n} binomial(3*n+k-1,n-k).

%H Seiichi Manyama, <a href="/A390337/b390337.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/(g * (1-3*x*g^2) * (1-x*g^4)) where g = 1+x*g^3 is the g.f. of A001764.

%F a(n) = Sum_{k=0..n} binomial(3*n-1,n-k) * Fibonacci(k+1).

%t Table[Sum[Binomial[3* n+k-1,n-k],{k,0,n}],{n,0,28}] (* _Vincenzo Librandi_, Jan 03 2026 *)

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

%o (Magma) [&+[Binomial(3*n+k-1, n-k): k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Jan 03 2026

%Y Cf. A088305, A279014, A390338, A390339.

%Y Cf. A038736, A038744, A390238.

%Y Cf. A000045, A001764.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 01 2025