A306760 -id:A306760 - 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.

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

Original entry on oeis.org

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

Vaclav Kotesovec, Mar 08 2019

			2.0344489454876164779803555318869026355979439863702376260005284165650078277571...

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

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

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

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}]

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

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).

A306789 a(n) = Product_{k=0..n} binomial(n + k, n).

Original entry on oeis.org

1, 2, 18, 800, 183750, 224042112, 1475939646720, 53195808994099200, 10587785727897772143750, 11721562427290210695200000000, 72596493516095364770534596279431168, 2527156530619699341247423878706695556300800, 496395279097923766533851314609410101501472675840000

Offset: 0

Vaclav Kotesovec, Mar 10 2019

nonn

Sum_{k=0..n} binomial(n + k, n) = binomial(2*n + 1, n).

Product_{k=1..n} binomial(k*n, n) = (n^2)! / (n!)^n.

Cf. A001700, A306760.

Table[Product[Binomial[n+k, n], {k, 0, n}], {n, 0, 13}]
Table[(n+1)^n * BarnesG[2*n+2] / (Gamma[n+2]^n * BarnesG[n+2]^2), {n, 0, 13}]

from math import factorial
from functools import lru_cache
@lru_cache(maxsize=None)
def A306789(n): return A306789(n-1)*2*n*factorial(2*n-1)**2//factorial(n)**3//n**(n-1) if n else 1 # Chai Wah Wu, Jun 26 2023

a(n) = (n+1)^n * BarnesG(2*n+2) / (Gamma(n+2)^n * BarnesG(n+2)^2).
a(n) ~ A * 2^(2*n^2 + 3*n/2 - 1/12) / (exp(n^2/2 + 1/6) * Pi^((n+1)/2) * n^(n/2 + 5/12)), where A is the Glaisher-Kinkelin constant A074962.
a(n) = a(n-1)*2n*(2n-1)!^2/(n!^3*n^(n-1)). - Chai Wah Wu, Jun 26 2023

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)).

A324589 a(n) = Product_{j=1..n, k=1..n} (1 + (j*k)^2).

Original entry on oeis.org

1, 2, 850, 9541930000, 62954953875193006250000, 2232026314050243695025069057306526600000000, 2378738322196706013428557679949358718247570924314917636028125000000000

Offset: 0

Vaclav Kotesovec, Mar 09 2019

nonn

Product_{j>=1, k>=1} (1 + 1/(j^3*k^3)) = 3.07044599622955113359633939413741321690850038945774000273914990604256664558...

Cf. A306760, A324590.

a:= n-> mul(mul((i*j)^2+1, i=1..n), j=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[Product[j^2*k^2 + 1, {j, 1, n}, {k, 1, n}], {n, 1, 8}]
Round[Table[Product[k^(1 + 2*n) * Gamma[1 - I/k + n] * Gamma[1 + I/k + n] * Sinh[Pi/k]/Pi, {k, 1, n}], {n, 1, 8}]]

a(n) ~ c * 4^n * Pi^(2*n) * n^(2*n*(2*n+1)) / exp(4*n^2), where c = 14.2467190172413789737182639605567415110439648274273645215657580983939589... = exp(1/3) * Product_{j>=1, k>=1} (1 + 1/(j^2*k^2)). - Vaclav Kotesovec, Mar 28 2019

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A324590 a(n) = n!^(4n) Product_{k=1..n} binomial(n + 1/k^2, n).

Original entry on oeis.org

1, 2, 1080, 16133644800, 139256878046022696960000, 6288402750181849898716908922601472000000000, 8322157105451357856813375261666887975745751468393837363200000000000000

Offset: 0

Vaclav Kotesovec, Mar 09 2019

nonn

Cf. A303670, A306760, A306774, A324589.

a:= n-> n!^(4*n)*mul(binomial(n+1/k^2, n), k=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[n!^(4*n) * Product[Binomial[1/k^2 + n, n], {k, 1, n}], {n, 1, 8}]

a(n) ~ n!^(4*n) * n^(Pi^2/6) / A303670.
a(n) ~ n^(4*n^2 + 2*n + Pi^2/6) * (2*Pi)^(2*n) / exp(4*n^2 - 1/3 - gamma*Pi^2/6 + c), where gamma is the Euler-Mascheroni constant A001620 and c = A306774 = Sum_{k>=2} (-1)^k * Zeta(k) * Zeta(2*k) / k.
a(n) = n!^n * A324596(n).

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A324596 a(n) = n!^(3n) Product_{k=1..n} binomial(n + 1/k^2, n).

Original entry on oeis.org

1, 2, 270, 74692800, 419731620267960000, 252716802910471719823692648960000, 59736659298524125157504488525739821430187940800000000, 16079377413231597423103950774423398920424350187193326745026311068057600000000000

Offset: 0

Vaclav Kotesovec, Mar 09 2019

nonn

Cf. A303670, A306760, A306774, A324589, A324597.

a:= n-> n!^(3*n)*mul(binomial(n+1/k^2, n), k=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[n!^(3*n) * Product[Binomial[n + 1/k^2, n], {k, 1, n}], {n, 1, 8}]

a(n) ~ n!^(3*n) * n^(Pi^2/6) / A303670.

a(n) ~ n^(3*n*(2*n+1)/2 + Pi^2/6) * (2*Pi)^(3*n/2) / exp(3*n^2 - 1/4 - gamma*Pi^2/6 + c), where gamma is the Euler-Mascheroni constant A001620 and c = A306774 = Sum_{k>=2} (-1)^k * Zeta(k) * Zeta(2*k) / k.

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A324597 a(n) = n!^(4n) Product_{k=1..n} binomial(n + 1/k^3, n).

Original entry on oeis.org

1, 2, 918, 11592504000, 86712397842439769400000, 3472997049383321958747830928094241894400000, 4152034082374349458781848863476555783741415883758270213129361920000000

Offset: 0

Vaclav Kotesovec, Mar 09 2019

nonn

In general, for m > 1, Product_{k=1..n} binomial(n + 1/k^m, n) ~ n^Zeta(m) / c(m), where c(m) = Product_{j>=1} Gamma(1 + 1/j^m).

Equivalently, c(m) = -gamma * Zeta(m) + Sum_{k>=2} (-1)^k*Zeta(k)*Zeta(m*k)/k, where gamma is the Euler-Mascheroni constant A001620.

Cf. A306760, A324596, A306778.

a:= n-> n!^(4*n)*mul(binomial(n+1/k^3, n), k=1..n):
seq(a(n), n=0..7);  # Alois P. Heinz, Jun 24 2023

Table[n!^(4*n) * Product[Binomial[n + 1/j^3, n], {j, 1, n}], {n, 1, 8}]

a(n) ~ n!^(4*n) * n^Zeta(3) / (Product_{j>=1} Gamma(1 + 1/j^3)).

a(n) ~ n^(4*n^2 + 2*n + Zeta(3)) * (2*Pi)^(2*n) / exp(4*n^2 - 1/3 - gamma*Zeta(3) + c), where c = A306778 = Sum_{k>=2} (-1)^k*Zeta(k)*Zeta(3*k)/k.

a(0)=1 prepended by Alois P. Heinz, Jun 24 2023

A325049 a(n) = Product_{i=0..n, j=0..n} (i!*j! + 1).

Original entry on oeis.org

2, 16, 6480, 97287175440, 1106928595945936328906250000, 856337316801926460412829104011102303451051923953906250000

Offset: 0

Vaclav Kotesovec, Mar 26 2019

nonn

Cf. A306760, A325051.

Table[Product[i!*j! + 1, {i, 0, n}, {j, 0, n}], {n, 0, 7}]
Table[BarnesG[n+2]^(2*n+2) * Product[1 + 1/(i!*j!), {i, 0, n}, {j, 0, n}], {n, 0, 7}]

from math import prod, factorial as f
def a(n): return prod(f(i)*f(j)+1 for i in range(n) for j in range(n))
print([a(n) for n in range(1, 7)]) # Michael S. Branicky, Feb 16 2021

a(n) ~ c * (2*Pi)^((n+1)^2) * n^((n+1)*(6*n^2 + 12*n + 5)/6) / (A^(2*n+2) * exp(3*n^3/2 + 7*n^2/2 + 11*n/6 - 1/3)), where c = Product_{i>=0, j>=0} (1 + 1/(i!*j!)) = 297.557220207478770881166673701943476275955597334672817171839377... and A is the Glaisher-Kinkelin constant A074962.

cp's OEIS Frontend

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

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

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A306789 a(n) = Product_{k=0..n} binomial(n + k, n).

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

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A324589 a(n) = Product_{j=1..n, k=1..n} (1 + (j*k)^2).

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A324590 a(n) = n!^(4*n) * Product_{k=1..n} binomial(n + 1/k^2, n).

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A324596 a(n) = n!^(3*n) * Product_{k=1..n} binomial(n + 1/k^2, n).

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A324597 a(n) = n!^(4*n) * Product_{k=1..n} binomial(n + 1/k^3, n).

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

A324590 a(n) = n!^(4n) Product_{k=1..n} binomial(n + 1/k^2, n).

A324596 a(n) = n!^(3n) Product_{k=1..n} binomial(n + 1/k^2, n).

A324597 a(n) = n!^(4n) Product_{k=1..n} binomial(n + 1/k^3, n).