A367331 - OEIS

%I #6 Nov 14 2023 17:26:57

%S 1,24,657,17664,477828,12888288,348197220,9397548288,253804616001,

%T 6851337236952,185014241769825,4994797849546752,134872057740184128,

%U 3641273395825798656,98320397048549301312,2654515896013953110016,71674988018612154171876

%N a(n) = 27^n * Sum_{k=0..n} (-1)^k*binomial(1/3, k)^2.

%C In general, for m>1, Sum_{k>=0} (-1)^k * binomial(1/m,k)^2 = 2^(1/m) * sqrt(Pi) / (Gamma(1 + 1/(2*m)) * Gamma(1/2 - 1/(2*m))).

%F a(n) ~  2^(5/3) * Pi * 3^(3*n + 1/2) / Gamma(1/3)^3.

%t Table[27^n*Sum[(-1)^k*Binomial[1/3, k]^2, {k, 0, n}], {n, 0, 16}]

%Y Cf. A358363, A367330, A367332, A367333.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Nov 14 2023