A372184 a(n) = 2^(1-3*n)*((2*n)!)^2/n.

1, 9, 675, 198450, 160744500, 291751267500, 1035425248357500, 6523179064652250000, 67867154988642009000000, 1102501932790489436205000000, 26741184379833321275152275000000, 933641711437500579000666529350000000, 45515033432578153226282493305812500000000

Offset

1,2

Links

Table of n, a(n) for n=1..13.

Formula

a(n) = Product_{k=1..n-1} A339483(k).

a(n) = A139757(n-1)*8^(1-n)*A134372(n-1).

a(n) = (2^n*n*2*Zeta_k(1-2*n)*Pi^(4*n))/(D_k^(2*n-1)*sqrt(D_k)*Zeta_k(2*n)) where Zeta_k() is the Dedekind zeta function over a real quadratic field with fundamental discriminant D_k = A003658(m) for some m > 1.

a(n) = 8^(1-n)*Integral_{x>=0} ( x^(2*n-(1/2))*BesselK(1, 2*sqrt(x)) ), where BesselK(m, ...) is the modified Bessel function K_m(...) of the first kind.

Sum_{n>=1} (x^(n-1)/a(n)) = (BesselI(1, 2*2^(3/4)*x^(1/4)) - BesselJ(1, 2*2^(3/4)*x^(1/4)))/(4*2^(1/4)*x^(3/4))

= (d/dx)((-2 + BesselI(0, 2*2^(3/4)*x^(1/4)) + BesselJ(0, 2*2^(3/4)*x^(1/4)))/4), where BesselI(m, ...) is the modified Bessel function I_m(...) of the first kind and BesselJ(m, ...) is the Bessel function J_m(...) of the first kind.

a(n) ~ Pi*2^(n+3)*exp(-4*n)*n^(4*n). - Stefano Spezia, Apr 22 2024

D-finite with recurrence 2*a(n) -n*(n-1)*(-1+2*n)^2*a(n-1)=0. - R. J. Mathar, May 20 2024

Prog

(PARI) a(n) = 2^(1-3*n)*(2*n)!^2/n

Crossrefs

Cf. A370411, A370412.

Cf. A003658, A139757, A134372, A339483.

Sequence in context: A054327 A209964 A287937 * A053973 A059492 A322488

Adjacent sequences: A372181 A372182 A372183 * A372185 A372186 A372187

Keyword

nonn

Author

Thomas Scheuerle, Apr 21 2024

Status

approved