OFFSET
0,2
FORMULA
a(n) = Product_{k=1..n}(2 - (2*k - 1))*Product_{k=1..n}(2 + (2*k - 1)).
a(n) = Sum_{k=0..n} A380570(n, k).
a(n) = (-(-1)^n*2^(-2*n - 1)*(2*n + 2)!*binomial(2*n, n))/((n + 1)*(2*n - 1)).
a(n) = -(-4)^n*Gamma(-1/2 + n)*Gamma(3/2 + n)/Pi.
a(n) = -(-1)^n*(2*n+1)!*Gamma(n - 1/2)/(2*sqrt(Pi)*n!).
a(n) = Integral_{u=0..1} Integral_{v=0..1} log(1/u)^(-3/2+n)*log(1/v)^(1/2+n) dvdu.
Sum_{k>=0} x^(2*k-1)/a(k) = -(Pi/2)*H_v(x), where H_v is the Struve Function at v = -2.
Sum_{k>=0} x^(2*k-1)/abs(a(k)) = (Pi/2)*L_v(x), where L_v is the modified Struve Function at v = -2.
a(n) = (n!)^2 [x^n] hypergeometric([-1/2, 3/2], [1], -4*x).
a(n) = -(-4)^n*Pochhammer(1/2, n - 1)*Pochhammer(1/2, n + 1). - Peter Luschny, Jan 29 2025
MAPLE
gf := hypergeom([-1/2, 3/2], [1], -4*x): ser := series(gf, x, 17):
seq((n!)^2*coeff(ser, x, n), n = 0..15); # Peter Luschny, Jan 29 2025
MATHEMATICA
Table[(-1)^n*Product[(2k+1)*(2k-3), {k, n}], {n, 0, 15}] (* James C. McMahon, Feb 02 2025 *)
PROG
(PARI) a(n) = Vec(hypergeom([-1/2, 3/2], 1, -4*x)+O(x^(n+1)))[n+1]*(n!)^2
(PARI) a(n) = prod(k=1, n, (2*k+1)*(2*k-3))*(-1)^n
(SageMath)
def a(n): return -(-4)**n*rising_factorial(1/2, n-1)*rising_factorial(1/2, n+1)
print([a(n) for n in range(16)]) # Peter Luschny, Jan 29 2025
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Thomas Scheuerle, Jan 28 2025
STATUS
approved