A307106 -id:A307106 - cp's OEIS frontend

A306769 Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^2 / k.

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

Offset: 1

Vaclav Kotesovec, Mar 09 2019

Sum_{k>=2} (-1)^k*Zeta(k)/k = A001620 (see MathWorld, formula 122).

			1.043402917574288733255289646671676030548470866046882561044570479769585...

Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A231132, A306760, A306765, A306774, A306778, A307106.

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

NSum[(-1)^k*Zeta[k]^2/k, {k, 2, Infinity}, WorkingPrecision -> 200, NSumTerms -> 100000]

sumalt(k=2, (-1)^k*zeta(k)^2/k) \\ Michel Marcus, Mar 09 2019

Equals log(A306765) + A001620^2.

A306907 a(n) = Product_{i=0..n, j=0..n, k=0..n} (ijk + 1).

Original entry on oeis.org

1, 2, 60750, 193002701276968128000000, 5076574217867350877310882935055477754989937924247841796875000000000000

Offset: 0

Vaclav Kotesovec, Mar 25 2019

nonn

Next term is too long to be included.

Cf. A306760, A307106.

Table[Product[i*j*k+1,{i,0,n},{j,0,n},{k,0,n}],{n,0,5}]

a(n) = (n!)^(3*n^2) * Product_{i=1..n, j=1..n, k=1..n} (1 + 1/(i*j*k)).
a(n) ~ exp(3*n^2*log(Gamma(n+1)) + (gamma + PolyGamma(0, n+1))^3 - c), where gamma is the Euler-Mascheroni constant A001620 and c = A307106 = Sum_{k>=2} (-1)^k * Zeta(k)^3 / k = 1.836921908595663783265640880112170343162564662453544904457...
a(n) ~ (2*Pi)^(3*n^2/2) * exp(-3*n^3 + n/4 + (log(n))^3 + 3*gamma*(log(n))^2 + gamma^3 - c) * n^(3*(n^3 + n^2/2 + gamma^2)).

cp's OEIS Frontend

A306769 Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^2 / k.

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A306907 a(n) = Product_{i=0..n, j=0..n, k=0..n} (ijk + 1).

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cp's OEIS Frontend

A306769 Decimal expansion of Sum_{k>=2} (-1)^k * Zeta(k)^2 / k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A306907 a(n) = Product_{i=0..n, j=0..n, k=0..n} (i*j*k + 1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A306907 a(n) = Product_{i=0..n, j=0..n, k=0..n} (ijk + 1).