A393244 - OEIS

%I #22 Feb 14 2026 20:40:17

%S 1,1,1,3,9,21,47,113,281,693,1701,4203,10455,26079,65157,163143,

%T 409401,1029249,2591453,6533759,16494509,41688825,105476031,267117585,

%U 677069575,1717578571,4360390711,11077347321,28159605693,71627334669,182295199467,464194851633

%N a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n,3*k).

%H Vincenzo Librandi, <a href="/A393244/b393244.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/sqrt((1-x)^2 - 4*x^3/(1-x)).

%F D-finite with recurrence (5 + 5*n)*a(n) + (-17 - 8*n)*a(n + 1) + (15 + 6*n)*a(n + 2) + (-13 - 4*n)*a(n + 3) + (n + 4)*a(n + 4) = 0. - _Robert Israel_, Feb 09 2026

%F a(n) ~ 5^(n + 1/2) / (2^(1/3) * sqrt(3*Pi*(21*2^(2/3) - 12*2^(1/3) - 11)*n) * (2^(4/3) - 2^(2/3) + 1)^(n-1)). - _Vaclav Kotesovec_, Feb 10 2026

%p f:= gfun:-rectoproc({(5 + 5*n)*a(n) + (-17 - 8*n)*a(n + 1) + (15 + 6*n)*a(n + 2) + (-13 - 4*n)*a(n + 3) + (n + 4)*a(n + 4), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 3},a(n),remember):

%p map(f, [$0..100]); # _Robert Israel_, Feb 09 2026

%t Table[Sum[Binomial[2*k,k]*Binomial[n,3*k],{k,0,Floor[n/3]}],{n,0,31}] (* _Vincenzo Librandi_, Feb 07 2026 *)

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

%o (Magma) [&+[Binomial(2*k, k)* Binomial(n, 3*k) : k in [0..Floor(n/3)]] : n in [0..35] ]; // _Vincenzo Librandi_, Feb 07 2026

%o (Python)

%o from math import comb

%o def A393244(n): return sum(comb(2*k, k) * comb(n, 3*k) for k in range(n // 3 + 1)) # _Aitzaz Imtiaz_, Feb 09 2026

%Y Cf. A002426, A026375, A393245.

%Y Cf. A098479, A217615.

%Y Cf. A360309.

%K nonn,easy

%O 0,4

%A _Seiichi Manyama_, Feb 07 2026