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.

1, 1, 2, 9, 38, 203, 1274, 9241, 76242, 705241, 7228724, 81323154, 996208647, 13199764580, 188096645269, 2868473840361, 46612699905873, 804069073376312, 14674260589117694, 282479516340515613, 5720210205895441171

Offset: 0

Author

Lechoslaw Ratajczak, Feb 17 2024

Keywords

• nonn

Comments

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,...

Examples

			a(0) = 1 because: floor(g(0 + 1/4)) = floor(g(1/4)) = floor(1) = 1.
a(1) = 1 because: floor(g(1 + 1/4)) = floor(g(5/4)) = floor(g(1/4)*(5/4)) = floor((1)*(5/4)) = 1.
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.
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.
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.
		

Crossrefs

Cf. A000302, A007696, A014513, A020087.

Programs

Maxima

makelist(floor(product(4*k+1, k, 1, n)/(4^n)), n, 0, 50);

Python

from math import prod
def A370397(n): return prod(range(5,(n<<2)+2,4))>>(n<<1) # Chai Wah Wu, Apr 28 2024

Formula

g(x) = Product_{k=0..floor(x - 1)} (x - k) for x >= 1.

g(x) = 1/(Product_{k=1..floor(-(x - 1))} (x + k)) for x < 0.

a(n) = floor((Product_{k=1..n} (4*k + 1))/4^n).

a(n) = floor(Product_{k=0..floor(n - 3/4)} (n - k + 1/4)).

a(n) = floor((4*n + 1)*gamma(n + 1/4)/gamma(1/4)).