A306765 Decimal expansion of lim_{k->oo} (k^A001620 / k!) * Product_{j=1..k} Gamma(1/j).

2, 0, 3, 4, 4, 4, 8, 9, 4, 5, 4, 8, 7, 6, 1, 6, 4, 7, 7, 9, 8, 0, 3, 5, 5, 5, 3, 1, 8, 8, 6, 9, 0, 2, 6, 3, 5, 5, 9, 7, 9, 4, 3, 9, 8, 6, 3, 7, 0, 2, 3, 7, 6, 2, 6, 0, 0, 0, 5, 2, 8, 4, 1, 6, 5, 6, 5, 0, 0, 7, 8, 2, 7, 7, 5, 7, 1, 1, 3, 2, 4, 4, 5, 0, 2, 6, 5, 0, 4, 0, 6, 1, 3, 5, 0, 7, 5, 0, 2, 9, 1, 2, 7, 1, 4

Offset

1,1

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..297

Formula

Equals exp(-gamma^2 + Sum_{j>=2} (-1)^j*Zeta(j)^2/j), where gamma is the Euler-Mascheroni constant A001620.

Equals exp(-gamma^2 + A306769).

Equals lim_{k->oo} k^(k*(2*k+1) + 2*gamma) * (2*Pi)^k * exp(1/6 + log(k)^2 - 2*k^2) / A306760(k).

Example

2.0344489454876164779803555318869026355979439863702376260005284165650078277571...

Maple

evalf(exp(-gamma^2 + Sum((-1)^j*Zeta(j)^2/j, j=2..infinity)), 100);

Mathematica

slogam = Table[Sum[LogGamma[1/j], {j, 1, n}], {n, 1, 1000}]; $MaxExtraPrecision = 1000; funs[n_] := E^slogam[[n]] * n^EulerGamma/n!; Do[Print[N[Sum[(-1)^(m + j) * funs[j*Floor[Length[slogam]/m]] * (j^(m - 1)/(j - 1)!/(m - j)!), {j, 1, m}], 80]], {m, 10, 100, 10}]

Prog

(PARI) exp(-Euler^2 + sumalt(j=2, (-1)^j*zeta(j)^2/j))

Crossrefs

Cf. A001620, A231132, A303670, A306760, A306769.

Sequence in context: A101336 A300001 A137218 * A087819 A290820 A278029

Adjacent sequences: A306762 A306763 A306764 * A306766 A306767 A306768

Keyword

nonn,cons

Author

Vaclav Kotesovec, Mar 08 2019

Status

approved