A386839 - OEIS

%I #29 Nov 09 2025 06:43:26

%S 1,5,31,201,1326,8824,59017,395986,2662707,17932178,120901309,

%T 815824636,5508658729,37214906598,251515412373,1700413287413,

%U 11499000357287,77778693896449,526186502987578,3560279119657039,24092560440190353,163052573806772721,1103598007689835900

%N a(n) = Sum_{k=0..n} binomial(3*n+2*k+1,n-k).

%H Vincenzo Librandi, <a href="/A386839/b386839.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: g/((1-3*x*g^2) * (1-x*g^5)) where g = 1+x*g^3 is the g.f. of A001764.

%F From _Vaclav Kotesovec_, Nov 09 2025: (Start)

%F Recurrence: 2*n*(2*n - 3)*(37*n^4 - 910*n^3 + 5747*n^2 - 13874*n + 11520)*a(n) = 3*(629*n^6 - 16580*n^5 + 130155*n^4 - 460480*n^3 + 808396*n^2 - 671280*n + 201600)*a(n-1) - 2*(2627*n^6 - 70049*n^5 + 588211*n^4 - 2299771*n^3 + 4624422*n^2 - 4624560*n + 1814400)*a(n-2) - (4847*n^6 - 128978*n^5 + 1070437*n^4 - 4117882*n^3 + 8116896*n^2 - 7926480*n + 3024000)*a(n-3) - 3*(3*n - 10)*(3*n - 8)*(37*n^4 - 762*n^3 + 3239*n^2 - 4962*n + 2520)*a(n-4).

%F a(n) ~ (31 + ((50933 - 4371*sqrt(93))/2)^(1/3) + (31*(1643 + 141*sqrt(93))/2)^(1/3)) * (2 + ((729 - 3*sqrt(93))^(1/3)/3 + (243 + sqrt(93))^(1/3)/3^(2/3))/2^(1/3))^n / 93 - 3^(3*n + 3/2) / (sqrt(Pi*n) * 2^(2*n-1)). (End)

%t Table[Sum[Binomial[3*n+2*k+1,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Nov 08 2025 *)

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

%o (Magma) [&+[Binomial(3*n+2*k+1, n-k): k in [0..n]] : n in [0..30] ]; // _Vincenzo Librandi_, Nov 08 2025

%Y Cf. A025174, A038744, A160906, A263134, A385250, A390452.

%Y Cf. A001764.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 06 2025