A191510 Product of terms in n-th row of A132818.

1, 9, 648, 360000, 1518750000, 48243443062500, 11480517255997440000, 20400479323264014247526400, 270090559531318533654528000000000, 26599911685677709861296622500000000000000, 19464564507161243794359748945629699456000000000000

Offset: 1

Author

Harlan J. Brothers, Jun 04 2011

Keywords

• easy
• nonn
• nice

Comments

Lim_{n -> inf} (a(n)*a(n+2))/a(n+1)^2 = e^2. Like A168510, this limit is asymptotic from above.

Examples

			For n=3, row 3 of A132818 = {6,18,6} and a(3)=648.
		

Links

• H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic constant e, Math. Intelligencer, Vol. 20, No. 4, (1998), 25-29.

Crossrefs

Cf. A132818, A002457. Related to e as in the cases of A168510 and A001142.

Programs

Mathematica

Table[Product[Product[((k + 1)/(k - 1))^k, {k, 2, j}], {j, 1, n}], {n, 1, 11}]
Table[(n + 1)^n * Hyperfactorial[n]^2 / (2^n * BarnesG[n+2]^2), {n, 1, 12}] (* Vaclav Kotesovec, Jul 11 2015 *)

Formula

a(n)=product[product[((k + 1)/(k - 1))^k, {k, 2, j}], {j, 1, n}].

a(n) ~ A^4 * exp(n^2 + 2*n + 5/6) / (n^(2/3) * 2^(2*n+1) * Pi^(n+1)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 11 2015