cp's OEIS Frontend

A370397 a(n) = floor(g(n + 1/4)), where g(x) is the function defined for all real numbers except the negative integers by the recurrence formula g(x+1) = g(x)*(x+1), with g(x) = 1 for 0 <= x < 1.

%I A370397 #17 Apr 28 2024 11:28:34
%S A370397 1,1,2,9,38,203,1274,9241,76242,705241,7228724,81323154,996208647,
%T A370397 13199764580,188096645269,2868473840361,46612699905873,
%U A370397 804069073376312,14674260589117694,282479516340515613,5720210205895441171
%N A370397 a(n) = floor(g(n + 1/4)), where g(x) is the function defined for all real numbers except the negative integers by the recurrence formula g(x+1) = g(x)*(x+1), with g(x) = 1 for 0 <= x < 1.
%C A370397 Connection between the gamma function and g(n): gamma(n + 1/k)/gamma(1/k) = g(n + 1/k)/(k*n + 1) for k=1,2,3,...
%F A370397 g(x) = Product_{k=0..floor(x - 1)} (x - k) for x >= 1.
%F A370397 g(x) = 1/(Product_{k=1..floor(-(x - 1))} (x + k)) for x < 0.
%F A370397 a(n) = floor((Product_{k=1..n} (4*k + 1))/4^n).
%F A370397 a(n) = floor(Product_{k=0..floor(n - 3/4)} (n - k + 1/4)).
%F A370397 a(n) = floor((4*n + 1)*gamma(n + 1/4)/gamma(1/4)).
%e A370397 a(0) = 1 because: floor(g(0 + 1/4)) = floor(g(1/4)) = floor(1) = 1.
%e A370397 a(1) = 1 because: floor(g(1 + 1/4)) = floor(g(5/4)) = floor(g(1/4)*(5/4)) = floor((1)*(5/4)) = 1.
%e A370397 a(2) = 2 because: floor(g(2 + 1/4)) = floor(g(9/4)) = floor(g(5/4)*(9/4)) = floor((5/4)*(9/4)) = 2.
%e A370397 a(3) = 9 because: floor(g(3 + 1/4)) = floor(g(13/4)) = floor(g(9/4)*(13/4)) = floor((5/4)*(9/4)*(13/4)) = 9.
%e A370397 a(4) = 38 because: floor(g(4 + 1/4)) = floor(g(17/4)) = floor(g(13/4)*(17/4)) = floor((5/4)*(9/4)*(13/4)*(17/4)) = 38.
%o A370397 (Maxima) makelist(floor(product(4*k+1, k, 1, n)/(4^n)), n, 0, 50);
%o A370397 (Python)
%o A370397 from math import prod
%o A370397 def A370397(n): return prod(range(5,(n<<2)+2,4))>>(n<<1) # _Chai Wah Wu_, Apr 28 2024
%Y A370397 Cf. A000302, A007696, A014513, A020087.
%K A370397 nonn
%O A370397 0,3
%A A370397 _Lechoslaw Ratajczak_, Feb 17 2024