A156648 -id:A156648 - cp's OEIS frontend

A375839 a(n) = Product_{k=0..n} (k^2 + n).

0, 2, 36, 1008, 41600, 2381400, 180457200, 17467670528, 2100621828096, 306960977700000, 53529274174376000, 10973787848179200000, 2611472797582941487104, 713649909809783275801472, 221870902844468552220000000, 77837994361783539267010560000

Offset: 0

Vaclav Kotesovec, Aug 31 2024

nonn

Cf. A156648, A101686.

Cf. A375840, A375841, A375842, A375843, A375844, A375845.

Table[Product[k^2 + n, {k, 0, n}], {n, 0, 15}]
Round[Table[Sqrt[n] * Gamma[1 - I*Sqrt[n] + n] * Gamma[1 + I*Sqrt[n] + n] * Sinh[Sqrt[n]*Pi] / Pi, {n, 0, 15}]]

a(n) ~ n^(2*n + 3/2) / exp(2*n - Pi*n^(1/2) + 1).

A326864 G.f.: Product_{k>=1} (1 + x^k/k^2) = Sum_{n>=0} a(n)*x^n/n!^2.

Original entry on oeis.org

1, 1, 1, 13, 100, 1876, 57636, 2051316, 104640768, 6819033600, 576652089600, 57187381536000, 7057192160793600, 1014733052692300800, 172646881540527744000, 33848454886497227289600, 7637231669166956976537600, 1948418678155880277481881600

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

			a(n) ~ c * (n-1)!^2, where c = A156648 = Product_{k>=1} (1 + 1/k^2) = sinh(Pi)/Pi = 3.67607791037497772...

Alois P. Heinz, Table of n, a(n) for n = 0..254

Cf. A007838, A249588, A326865.

Cf. A087132.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 27 2023

nmax = 20; CoefficientList[Series[Product[(1+x^k/k^2), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!^2

A334401 Decimal expansion of sinh(Pi).

Original entry on oeis.org

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

Offset: 2

Ilya Gutkovskiy, Apr 26 2020

This constant is transcendental.

			(e^Pi - e^(-Pi))/2 = 11.5487393572577483779773343153884...

Index entries for transcendental numbers

Cf. A000796, A001113, A039661, A068470, A073742, A093580, A156648, A323983, A334402.

Mathematica
```
RealDigits[Sinh[Pi], 10, 110] [[1]]
```

Equals Sum_{k>=0} Pi^(2*k+1)/(2*k+1)!.

Equals 2 * Product_{k>=1} (4*k^2+4)/(4*k^2-1).

A307216 Decimal expansion of Product_{k>=1} (1 + 1/k^5).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Mar 29 2019

			2.07422504479637891390708968594384056977125337962227288334734036988361960596259...

Eric Weisstein's World of Mathematics, Infinite Product.

Cf. A156648, A073017, A258870, A258871.

evalf(Product(1 + 1/j^5, j = 1..infinity), 120);

RealDigits[Chop[N[Product[(1 + 1/n^5), {n, 1, Infinity}], 120]]][[1]]
With[{g = GoldenRatio}, Chop[N[1/(Gamma[1/(2*g^2) - I*5^(1/4)/(2*Sqrt[g])] * Gamma[g^2/2 + I*5^(1/4) * Sqrt[g]/2] * Gamma[g^2/2 - I*5^(1/4) * Sqrt[g]/2] * Gamma[1/(2*g^2) + I*5^(1/4)/(2*Sqrt[g])]), 120]]]
N[1/Abs[Gamma[Exp[2*Pi*I/5]]*Gamma[Exp[6*Pi*I/5]]]^2, 120] (* Vaclav Kotesovec, Apr 27 2020 *)

default(realprecision, 120); exp(sumalt(j=1, -(-1)^j*zeta(5*j)/j))

Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(5*j)/j)).

Equals 1/(Gamma(1/(2*phi^2) - i*(5^(1/4)/(2*sqrt(phi)))) * Gamma(phi^2/2 + i*5^(1/4)*(sqrt(phi)/2)) * Gamma(phi^2/2 - i*5^(1/4)*(sqrt(phi)/2)) * Gamma(1/(2*phi^2) + i*(5^(1/4)/(2*sqrt(phi))))), where i is the imaginary unit and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A308717 Decimal expansion of cosh(sqrt(3)Pi/2)sech(Pi/2).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Jun 19 2019

			3.040191551642553075601860027900757233823854905879...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A001844, A156648, A308642, A308718.

RealDigits[Cosh[Sqrt[3] Pi/2] Sech[Pi/2], 10, 120][[1]]

cosh(sqrt(3)*Pi/2)/cosh(Pi/2) \\ Michel Marcus, Jun 20 2019

Equals Product_{k>=0} (1 + 1/(2*k*(k + 1) + 1)).

Equals Product_{k>=0} (1 + 1/A001844(k)).

A330864 Decimal expansion of sinh(Pi/2)/2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 28 2020

This constant is transcendental.

			(1 + 1/2^2) * (1 - 1/3^2) * (1 + 1/4^2) * (1 - 1/5^2) * (1 + 1/6^2) * ... = (e^(Pi/2) - e^(-Pi/2))/4 = 1.15064945115364743673152001...

Index entries for transcendental numbers

Cf. A042972, A049006, A090986, A156648, A175615, A308715, A308716, A308718, A323983, A330865, A334401.

Mathematica
```
RealDigits[Sinh[Pi/2]/2, 10, 110] [[1]]
```

sinh(Pi/2)/2 \\ Michel Marcus, Apr 28 2020

Equals Sum_{k>=1} Pi^(2*k-1)/(4^k*(2*k-1)!).
Equals Product_{k>=2} (1 + (-1)^k/k^2).
Equals (i^(-i) - i^i)/4, where i is the imaginary unit.

A330865 Decimal expansion of cosh(Pi/2)/Pi.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 28 2020

			(1 - 1/2^2) * (1 + 1/3^2) * (1 - 1/4^2) * (1 + 1/5^2) * (1 - 1/6^2) * ... = (e^(Pi/2) + e^(-Pi/2))/(2*Pi) = 0.7986963159564630848638067...

Cf. A042972, A049006, A049541, A156648, A175615, A308715, A308716, A308718, A323982, A330864, A334402.

RealDigits[Cosh[Pi/2]/Pi, 10, 110] [[1]]

cosh(Pi/2)/Pi \\ Michel Marcus, Apr 28 2020

Equals Sum_{k>=0} Pi^(2*k-1)/(4^k*(2*k)!).
Equals Product_{k>=2} (1 - (-1)^k/k^2).
Equals (i^(-i) + i^i)/(2*Pi), where i is the imaginary unit.

cp's OEIS Frontend

A375839 a(n) = Product_{k=0..n} (k^2 + n).

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A326864 G.f.: Product_{k>=1} (1 + x^k/k^2) = Sum_{n>=0} a(n)*x^n/n!^2.

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A334401 Decimal expansion of sinh(Pi).

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A307216 Decimal expansion of Product_{k>=1} (1 + 1/k^5).

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A308717 Decimal expansion of cosh(sqrt(3)*Pi/2)*sech(Pi/2).

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A330864 Decimal expansion of sinh(Pi/2)/2.

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A330865 Decimal expansion of cosh(Pi/2)/Pi.

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A308717 Decimal expansion of cosh(sqrt(3)Pi/2)sech(Pi/2).