A342099 Product of first n tangent numbers.

1, 1, 2, 32, 8704, 69074944, 24438162587648, 546639076930132901888, 1040668139730671025101058605056, 218400176068773166949459169210753567686656, 6353017630286823410670432558608528274164598967780769792

Offset

0,3

Links

Table of n, a(n) for n=0..10.

Eric Weisstein's World of Mathematics, Tangent Number.

Formula

a(n) = Product_{k=1..n} A000182(k).

a(n) ~ c * 2^(n*(2*n+3)) * n^(n^2 + n/2 - 1/24) / (Pi^(n*(2*n+1)/2) * exp(n*(3*n+1)/2)), where c = 1.3336306469174300191610203408604845574627820502002809243182947395752927990...

Maple

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= proc(n) a(n):=`if`(n=0, 1, a(n-1)*b(2*n-1, 0)) end:
seq(a(n), n=0..12);  # Alois P. Heinz, Mar 05 2021

Mathematica

Table[Product[(-1)^k * (16^k - 4^k)*Zeta[1 - 2*k], {k, 1, n}], {n, 0, 12}]
Table[Product[2*PolyGamma[2*k-1, 1/2]/Pi^(2*k), {k, 1, n}], {n, 0, 12}]
FoldList[Times, 1, Table[(-1)^n * (16^n - 4^n)*Zeta[1 - 2*n], {n, 1, 12}]]

Prog

(Python)
from math import prod
from sympy import bernoulli
def A342099(n): return abs(prod(((2-(2<<(m:=i<<1)))*bernoulli(m)<<m-2)//i for i in range(1, n+1))) # Chai Wah Wu, Apr 16 2023

Crossrefs

Cf. A000182.

Sequence in context: A202629 A129349 A180127 * A091804 A012853 A128146

Adjacent sequences: A342096 A342097 A342098 * A342100 A342101 A342102

Keyword

nonn

Author

Vaclav Kotesovec, Mar 05 2021

Status

approved